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New Volume Comparison Results and Volume Growth Rigidity of Gradient Ricci Almost Solitons

Published 9 Oct 2023 in math.DG | (2310.05583v4)

Abstract: In this paper, we establish a new volume comparison theorem for a complete manifold with a function $\rho(x)$ as the lower bound of the Bakry-Emery Ricci curvature. As applications, we obtain a new volume rigidity result of the gradient Ricci almost solitons. Furthermore, we extend the results of Cao and Zhou \cite{CZ} to shrinking gradient Ricci almost solitons and get the rigidity result with respect to the maximal volume growth.

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  1. Barros, A., Batista, R., Ribeiro Jr, E. (2014). Compact almost Ricci solitons with constant scalar curvature are gradient. Monatsh. Math., 174, 29–39. [4] Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C., Mazzieri, L. (2017). The Ricci-Bourguignon flow. Pacific J. Math., 287(2), 337–370. [5] Catino, G., Mazzieri, L. (2016). Gradient Einstein solitons. Nonlinear Anal., 132, 66–94. [6] Cao, H.-D., Zhou, D. (2010). On complete gradient shrinking Ricci solitons. J. Differential Geom., 85(2), 175-186. [7] Cao, H.-D., Chen, B.-L., Zhu, X.-P. (2008). Recent Developments on Hamilton’s Ricci flow. Surveys in Differential Geometry XII. [8] Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Catino, G., Cremaschi, L., Djadli, Z., Mantegazza, C., Mazzieri, L. (2017). The Ricci-Bourguignon flow. Pacific J. Math., 287(2), 337–370. [5] Catino, G., Mazzieri, L. (2016). Gradient Einstein solitons. Nonlinear Anal., 132, 66–94. [6] Cao, H.-D., Zhou, D. (2010). On complete gradient shrinking Ricci solitons. J. Differential Geom., 85(2), 175-186. [7] Cao, H.-D., Chen, B.-L., Zhu, X.-P. (2008). Recent Developments on Hamilton’s Ricci flow. Surveys in Differential Geometry XII. [8] Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Catino, G., Mazzieri, L. (2016). Gradient Einstein solitons. Nonlinear Anal., 132, 66–94. [6] Cao, H.-D., Zhou, D. (2010). On complete gradient shrinking Ricci solitons. J. Differential Geom., 85(2), 175-186. [7] Cao, H.-D., Chen, B.-L., Zhu, X.-P. (2008). Recent Developments on Hamilton’s Ricci flow. Surveys in Differential Geometry XII. [8] Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Cao, H.-D., Zhou, D. (2010). On complete gradient shrinking Ricci solitons. J. Differential Geom., 85(2), 175-186. [7] Cao, H.-D., Chen, B.-L., Zhu, X.-P. (2008). Recent Developments on Hamilton’s Ricci flow. Surveys in Differential Geometry XII. [8] Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Cao, H.-D., Chen, B.-L., Zhu, X.-P. (2008). Recent Developments on Hamilton’s Ricci flow. Surveys in Differential Geometry XII. [8] Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). 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Math., 352, 1096-1154. Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. 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[13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. 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Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Cheng, X., Ribeiro Jr, E., Zhou, D. (2022). Volume growth estimates for Ricci solitons and quasi-Einstein manifolds. J. Geom. Anal., 32(2), 62. [9] Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. 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Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). 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New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. 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[15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). 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Math., 352, 1096-1154. Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). 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Math., 352, 1096-1154. Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). 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[23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
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  7. Feitosa, F. E. S., Freitas Filho, A. A., Gomes, J. N. V., Pina, R. S. (2019). Gradient Ricci almost soliton warped product. J. Geom. Phys., 143, 22–32. [10] Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). 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Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). 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[13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. 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Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  8. Hamilton, R. S. (1995). The formation of singularities in the Ricci flow. Surveys in Differential Geometry (Cambridge, MA, 1993), 2, 7-136. [11] Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). 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(2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  9. Hamilton, R. S. (1988). The Ricci flow on surfaces. Contemp. Math., 71, 237-261. [12] Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Li, P. (2012). Geometric analysis. Cambridge Studies in Advanced Mathematics, 134. Cambridge University Press. [13] Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. [14] Ni, L., Wallach, N. (2008). On a classification of gradient shrinking solitons. Math. Res. Lett., 15(5), 941-955. [15] Perelmann, G. (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159. [16] Perelmann, G. (2003). Ricci flow with surgery on three manifolds. arXiv:math.DG/0303109. [17] Petersen, P. (2006). Riemannian geometry. New York: Springer. [18] Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Naber, A. (2010). Noncompact shrinking four solitons with nonnegative curvature. J. Reine Angew. Math., 645, 125-153. 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  16. Petersen, P., Wylie, W. (2010). On the classification of gradient Ricci solitons. Geom. Topol., 14, 2277–2300. [19] Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  17. Petersen, P., Wylie, W. (2009). Rigidity of gradient Ricci solitons. Pacific J. Math., 241, 329–345. [20] Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  18. Pigola, S., Rigoli, M., Rimoldi, M., Setti, A. G. (2011). Ricci almost solitons. Ann. Sci. Norm. Super. Pisa-Cl. Sci., 10(4), 757-799. [21] Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  19. Wei, G., Wylie, W. (2009). Comparison geometry for the Bakry-Emery Ricci tensor. J. Differential Geom., 83, 377–405. [22] Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  20. Zhang, Z. H. (2009). On the completeness of gradient Ricci solitons. Proc. Amer. Math. Soc., 137, 2755–2759. [23] Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.
  21. Zhang, Q. S., Zhu, M. (2019). New volume comparison results and applications to degeneration of Riemannian metrics. Adv. Math., 352, 1096-1154.

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