Sormani–Wenger Intrinsic Flat Convergence

• SWIF convergence is a notion that defines convergence of oriented metric spaces by embedding them into a common metric space and minimizing the filling volume between pushforward currents.
• It overcomes limitations of Gromov–Hausdorff convergence by effectively handling singular sets and mass cancellations with precise volume controls.
• SWIF techniques are applied in geometric analysis, including studies of Ricci flow, Calabi–Yau transitions, and minimal surfaces, ensuring robust comparison of degenerate manifolds.

The Sormani–Wenger Intrinsic Flat (SWIF) convergence is a geometric-analytic notion designed to address the convergence of sequences of oriented metric spaces, especially integral current spaces arising from Riemannian manifolds, in the presence of degenerations, singularities, or potential “loss of mass.” Developed in the context of overcoming limitations of Gromov–Hausdorff convergence, the SWIF distance measures how close two such spaces are by embedding them into a common metric space and quantifying the minimal “filling volume” between pushforward currents. This framework is robust under singularity formation and allows for precise comparison of spaces even when mass collapses or cancels. The theory has been instrumental in progress on geometric flows, degeneration of metrics, metric geometry of singular spaces, and applications in mathematical physics such as extremal transitions of Calabi–Yau spaces.

1. Definition of SWIF Distance and Convergence

The Sormani–Wenger Intrinsic Flat distance, denoted $d_{\mathcal{F}}$, is defined between two integral current spaces by first isometrically embedding them into a common complete metric space $Z$ and then computing the flat distance between the push-forward integral currents. If $(M_1, d_1, T_1)$ and $(M_2, d_2, T_2)$ are oriented Riemannian manifolds considered as integral current spaces (following Ambrosio–Kirchheim), the SWIF distance is given by

$$d_{\mathcal{F}}(M_1, M_2) := \inf_{\varphi_i, Z} d_F^Z(\varphi_{1\#}T_1,\, \varphi_{2\#}T_2)$$

where the infimum is over all isometric embeddings $\varphi_i : M_i \to Z$ and $d_F^Z$ is the classical flat distance. Key features:

• The flat distance itself is

$$d_F^Z(A, B) = \inf \{ \mathbf{M}(U) + \mathbf{M}(V)\, :\, A - B = U + \partial V \}$$

where $U$ (an $m$-current) and $V$ (an $(m+1)$-current) live in $Z$, and $\mathbf{M}$ is the mass.

• Only the “settled completion” (points with positive mass density) are represented in the limit.
• $d_{\mathcal{F}}$ is zero if and only if the two integral current spaces are current-preserving isometric.

SWIF convergence occurs if $d_{\mathcal{F}}(M_i, M_\infty) \to 0$ as $i\to\infty$.

2. Geometric Setting and Structural Assumptions

The main analytic results in (Lakzian, 2012) establish SWIF convergence for families of metrics $g_i$ on a fixed compact Riemannian $n$-manifold $M$ with particular control near singular sets $S$. The principal geometric hypotheses include:

• Smooth Convergence Away from $S$: $g_i \to g_\infty$ smoothly on $M \setminus S$.
• Singular Set $S$: $S$ is closed, with $\mathcal{H}^{n-1}(S) = 0$; in prior work, $S$ was assumed to be a codimension-2 submanifold.
• Exhaustion $W_j$: A sequence $W_j$ providing a connected, precompact exhaustion of $M \setminus S$.

Quantitative uniform bounds are imposed on $W_j$:

• Extrinsic Diameter Bound: $\diam_{g_i}(W_j) \leq D_0$.
• Boundary Volume Bound: $\vol_{g_i}(\partial W_j) \leq A_0$.
• Residual Volume Bound: $\vol_{g_i}(M \setminus W_j) \leq V_j$ with $\lim_{j \to \infty} V_j = 0$.

Under these conditions, the following theorems are established:

• If $g_i$ admit a uniform lower Ricci curvature bound, or a uniform linear contractibility function, and converge smoothly away from a singular set with $\mathcal{H}^{n-1}(S) = 0$, then the Gromov–Hausdorff limit agrees with the metric completion of $(M^n \setminus S, g_\infty)$.
• Even without Ricci or contractibility bounds, one obtains

$\lim_{j \to \infty} d_{\mathcal{F}}(M'_j, N') = 0,$

where $M'_j$ and $N'$ are the settled completions of $(M, g_j)$ and $(M_\infty \setminus S, g_\infty)$.

3. Mathematical Estimates and Technical Formulas

A central mathematical estimate for the SWIF distance in the presence of controlled geometry is: $d_{\mathcal{F}}(M_1, M_2) \leq (2\hat{h} + a) \left( \Vol_m(U_1) + \Vol_m(U_2) + \Vol_{m-1}(\partial U_1) + \Vol_{m-1}(\partial U_2) \right) + \Vol_m(M_1 \setminus U_1) + \Vol_m(M_2 \setminus U_2),$ where $a$ and $\hat{h}$ are parameters from comparison of the metrics, and $U_j$ are subsets realizing volume and diameter controls.

Further, the extrinsic diameter and boundary volume bounds ensure that no “wild” metric distortions or volume concentrations are lost in the limit, and the smallness of $\Vol(M^n \setminus W_j)$ serves to control the “edge effect” at the singular set.

4. Illustrative Examples and Applications

The necessity of each hypothesis is supported by explicit construction:

• Pinched Torus: A sequence of metrics on $S^1 \times S^1$ pinched along $S=\{0\} \times S^1$, smoothly converging away from $S$, with the limiting metric completion sewing together two ends in a topologically nontrivial way.
• Slit and Pulled Tori: Removing the uniform “well-embeddedness” or substituting the extrinsic diameter with the intrinsic version potentially causes the SWIF limit to differ from the expected metric completion.
• Scalar Curvature and Minimal Surfaces: In three dimensions, examples with unbounded diameters or only scalar (not Ricci) curvature bounds show failure of SWIF or GH convergence, demonstrating the necessity of the more restrictive geometric assumptions.

Principal application: The main theorems are applied to the analytic context of Calabi–Yau conifolds, verifying an “analytic soft” version of the Candelas–de la Ossa conjecture. For Ricci-flat Kähler metrics on smooth resolutions/smoothings, SWIF and GH convergence to the metric completion of the smooth region is established, confirming continuity of extremal transitions and flops.

5. Comparative and Theoretical Analysis

Earlier work required the singular set $S$ to be a smooth submanifold of codimension two. The results in (Lakzian, 2012) generalize this to any closed $S$ with $\mathcal{H}^{n-1}(S)=0$, and even beyond by reducing the measure condition to extrinsic diameter estimates on $\partial W_j$. The SWIF framework distinguishes itself from metric-only convergence (e.g., Gromov–Hausdorff) by being sensitive to measure “cancellations” and by collapsing sets of zero lower mass density (“tip” points).

Notably, when a uniform linear contractibility function or a lower Ricci bound is present, the GH and SWIF limits coincide; without such, cancellation and loss of “mass” not visible in the metric completion can occur.

6. Broader Significance and Directions for Further Investigation

SWIF convergence is robust under topology or smooth-structure change. It is particularly potent for studying sequences of manifolds in contexts such as Ricci flow through singularities, collapsing geometries, and spaces arising in minimal surface and Kähler geometry. The main theorems guarantee that with suitable diameter, boundary volume, and edge volume control, as well as curvature or contractibility bounds, intrinsic flat limits agree with metric completions of the smooth complement of the singular set.

Implications include:

• A framework for establishing geometric and analytic stability across singular transitions in geometrically rich settings (e.g., Calabi–Yau transitions).
• Potential for further weakening geometric conditions; e.g., relaxing the dimension or regularity requirements on the singular set, or replacing curvature by contractibility or weaker integral bounds.
• Promising avenues for a deeper understanding of the relationship and differences between GH and SWIF convergence, especially in the paper of spaces with singularities and in formulating generalized scalar curvature bounds for metric limit spaces.

In conclusion, SWIF convergence, as refined and applied in (Lakzian, 2012), provides a technically powerful and flexible tool for studying the convergence of manifolds and integral current spaces in the presence of singularities, with precise quantitative hypotheses ensuring agreement with metric completions and illustrating the pivotal role of geometric and measure-theoretic controls.