Revised Cone Model in Synthetic Ricci Geometry

• Revised Cone Model is a framework that transfers generalized Ricci curvature bounds to singular cone spaces using the curvature–dimension condition (CD(K,N)).
• It employs optimal transport and synthetic geometry to extend classical results, like Bishop–Gromov and Bonnet–Myers, to non-smooth metric measure spaces.
• The model facilitates practical insights into spectral gap estimates, eigenvalue bounds, and the geometric analysis of manifolds with lower Ricci curvature.

The revised cone model synthesizes key advances in the geometric and analytic characterization of cones associated with Riemannian manifolds, with particular emphasis on the transfer and generalization of Ricci curvature lower bounds via the curvature–dimension condition (CD(K,N)). This framework has profound implications for the study of metric measure spaces, spectral gaps, and the analysis of singular spaces arising as limits of manifolds with lower Ricci curvature bounds. Below, the major aspects of the revised cone model are delineated in detail.

1. Generalized Ricci Bounds in Cone Constructions

The central insight of the revised cone model is that generalized Ricci lower bounds, formalized via the CD(K,N) condition (where $K$ is the lower bound on Ricci curvature and $N$ the dimension parameter), can be systematically transferred from a base space to both Euclidean and spherical cones over it. If an $n$-dimensional weighted Riemannian manifold $M$ satisfies $\operatorname{Ric}^{(N)} \geq N-1$ (or equivalently the curvature–dimension condition $\mathrm{CD}(N-1, N)$), then:

• Its $N$-Euclidean cone $\mathrm{Con}(M)$ satisfies $\mathrm{CD}(0, N+1)$; in particular, when $N = n$, this yields nonnegative Ricci curvature in the cone.
• Its $N$-spherical cone $\Sigma(M)$ satisfies $\mathrm{CD}(N, N+1)$, inheriting a positive lower curvature bound in the metric measure sense.

This transfer is robust even though the cones typically possess singularities at their apices or poles and are not smooth manifolds. Thus, the model provides crucial new tools for the study of singular spaces, especially in the context of metric measure geometry.

2. Curvature–Dimension Conditions and Synthetic Geometry

The curvature–dimension condition $\mathrm{CD}(K, N)$ replaces traditional Ricci lower bounds with a requirement on convexity of entropy functionals along Wasserstein geodesics in the space of probability measures. In the optimal transport formulation:

$$\mathcal{S}_{N'}(\nu_t \mid \mathfrak{m}) = -\int_M \rho_t^{1 - 1/N'} \, dm$$

is convex along geodesics, where $\rho_t$ is the interpolated measure density. For nonbranching spaces, a pointwise inequality is satisfied:

$$\rho_t^{-1/N}(\gamma(t)) \geq \tau_{K, N}^{(1-t)}(d(\gamma(0), \gamma(1))) \rho_0^{-1/N}(\gamma(0)) + \tau_{K,N}^{(t)}(d(\gamma(0), \gamma(1))) \rho_1^{-1/N}(\gamma(1))$$

where $\tau_{K,N}^{(t)}$ are volume distortion coefficients. In cones, the transformation systematically shifts the CD parameter pair, enabling direct computation and verification of synthetic curvature properties on the higher-dimensional, singular-cone spaces.

3. Metric Measure Spaces and Singularities

The revised model treats Euclidean and spherical cones as metric measure spaces, accommodating the breakdown of classical differentiable structures at singularities (the cone tip for Euclidean cones, poles for spherical cones). The Euclidean cone,

$$\mathrm{Con}(M) = (M \times [0, \infty))/ (M \times \{0\})$$

admits the metric

$$d_{\mathrm{Con}}((x,s),(y,t)) = \sqrt{s^2 + t^2 - 2st\cos(d(x,y) \wedge \pi)}$$

and the measure

$dm_N(x,s) = dm(x) \otimes s^N ds$

Similarly, the spherical cone metric is specified by the cosine law and integrated with a measure of the form $dm(x) \otimes \sin^N s ds$. The metric measure approach, rooted in optimal transport and synthetic Ricci bounds, is robust to singularities and enables the extension of “Ricci curvature” to spaces well beyond classical Riemannian geometry.

4. Equivalence of Curvature–Dimension Conditions

A foundational result is the equivalence:

• $(M, d, m)$ satisfies $\mathrm{CD}(N-1, N)$ if and only if its $N$-Euclidean cone satisfies $\mathrm{CD}(0, N+1)$ and its $N$-spherical cone satisfies $\mathrm{CD}(N, N+1)$.

Proof proceeds in two steps:

• Optimal transport on the cone avoids the singularity, ensuring almost no mass is transferred through the tip or poles.
• On the punctured cone (with the singularity removed), one expresses the generalized Ricci curvature via classical formulas:

$$\operatorname{Ric}^{(N+1, W)}_{(x, r)}((v, t),(v, t)) = \operatorname{Ric}^{(N, V)}_x(v, v) - (N-1)\|v\|^2$$

This construction ensures the CD condition genuinely "lifts" across the cone operation, precisely reflecting the curvature–dimension properties of the base manifold in the higher-dimensional cone.

5. Mathematical Formulations and Key Equations

Core equations in the revised cone model include:

• Euclidean cone metric: $$d_{\mathrm{Con}}((x,s),(y,t)) = \sqrt{s^2 + t^2 - 2st\cos(d(x,y) \wedge \pi)}$$
• Spherical cone metric: $$\cos(d_\Sigma((x,s),(y,t))) = \cos s \cos t + \sin s \sin t \cos(d(x,y) \wedge \pi)$$
• Measures: $dm_N(x,s) = dm(x) \otimes s^N ds$, $d\hat{m}_N(x,s) = dm(x) \otimes \sin^N s ds$
• CD condition (nonbranching case):

$$\rho_t^{-1/N}(\gamma(t)) \ge \tau_{K,N}^{(1-t)}(d(\gamma(0),\gamma(1)))\rho_0^{-1/N}(\gamma(0)) + \tau_{K,N}^{(t)}(d(\gamma(0),\gamma(1)))\rho_1^{-1/N}(\gamma(1))$$

These formulas govern the curvature–dimension correspondences under the cone transformation and allow the direct verification of Ricci bounds in the synthetic setting.

6. Applications, Consequences, and Analytic Implications

The revised cone model has several significant implications:

• It generalizes, via synthetic Ricci curvature, classical theorems including Brunn–Minkowski, Bishop–Gromov, Bonnet–Myers, and Lichnerowicz, to settings that include singular spaces constructed by cone operations.
• It yields sharp spectral gap estimates: for an $n$-spherical cone over $M$ (with $\operatorname{Ric}\geq n-1$ and $\operatorname{diam}(M)\leq\pi$):

$$\int_{\Sigma(M)} f^2 d\hat{m}_n \leq \frac{1}{n+1} \int_{\Sigma(M)} |\nabla f|^2 d\hat{m}_n$$

implying a first nonzero Laplacian eigenvalue $\lambda_1 \geq n+1$.

• It applies in the analysis of limits (e.g., Gromov–Hausdorff limits) of manifolds with Ricci lower bounds, as tangent cones in the limiting spaces are cones as described.
• The model enables extension of Ricci curvature notions, via synthetic geometric methods, to even more singular spaces such as hyperbolic cones and $(\kappa,N)$-cones.

7. Broader Impact in Geometric Analysis

The revised cone model fundamentally expands the analytic and geometric toolkit for understanding spaces that are neither manifolds nor Alexandrov spaces, establishing a direct correspondence between curvature–dimension properties of base spaces and their associated cones. It enables rigorous analytic results—including isoperimetric inequalities and spectral gap bounds—on spaces with singularities, illuminating the structure of metric measure limits and offering new approaches for synthetic Ricci curvature in highly generalized contexts.

In summary, the revised cone model not only systematically lifts important curvature properties from smooth manifolds to non-smooth, singular cones, but establishes analytic equivalence via the curvature–dimension condition. This provides foundational support for geometric and analytic results on singular metric measure spaces and opens further directions for the study of curvature and spectrum in the synthetic geometry setting.