Curvature of sub-Riemannian spaces

Continuity of the isoperimetric profile of a complete Riemannian manifold under sectional curvature conditions

Let $M$ be a complete Riemannian manifold possessing a strictly convex Lipschitz continuous exhaustion function. We show that the isoperimetric profile of $M$ is a continuous and non-decreasing function. Particular cases are Hadamard manifolds and complete non-compact manifolds with strictly positive sectional curvatures.

Curvature Measures of Pseudo-Riemannian Manifolds

The Weyl principle is extended from the Riemannian to the pseudo-Riemannian setting, and subsequently to manifolds equipped with generic symmetric $(0,2)$-tensors. More precisely, we construct a family of generalized curvature measures attached to such manifolds, extending the Riemannian Lipschitz-Killing curvature measures introduced by Federer. We then show that they behave naturally under isometric immersions, in particular they do not depend on the ambient signature. Consequently, we extend Theorema Egregium to surfaces equipped with a generic metric of changing signature, and more general…

具有 Hölder 连续曲率的 Riemann 曲面上距离函数的有限性原理

A finiteness principle for distance functions on Riemannian surfaces with Hölder continuous curvature

本文研究 Hölder 连续 Gauss 曲率 Riemann 曲面上测地线到点的距离函数，证明类似 Whitney 延拓理论的有限性原理，并将其用于带曲率半范数控制的等距嵌入。

一致曲率与可收缩性约束下黎曼2-流形的收敛性

Convergence of Riemannian 2-manifolds under a uniform curvature and contractibility bound

该研究针对满足同胚、总绝对曲率一致有界的一致半局部1-连通闭黎曼2-流形序列，刻画其Gromov-Hausdorff极限，推广了Burago和Shioya的相关研究。

Volume of small balls and sub-Riemannian curvature in 3D contact manifolds

We compute the asymptotic expansion of the volume of small sub-Riemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structure

On sub-Riemannian geodesic curvature in dimension three

We introduce a notion of geodesic curvature $k_ζ$ for a smooth horizontal curve $ζ$ in a three-dimensional contact sub-Riemannian manifold, measuring how much a horizontal curve is far from being a geodesic. We show that the geodesic curvature appears as the first corrective term in the Taylor expansion of the sub-Riemannian distance between two points on a unit speed horizontal curve $$ d_{SR}^2( ζ(t),ζ(t+ε))=ε^2-\frac{k_ζ^2(t)}{720} ε^6 +o(ε^{6}). $$ The sub-Riemannian distance is not smooth on the diagonal, hence the result contains the existence of such an asymptotics. This can be seen as …

统一的合成里奇曲率下界用于黎曼和亚黎曼结构

Unified synthetic Ricci curvature lower bounds for Riemannian and sub-Riemannian structures

本文提出通过研究度量测度空间，统一黎曼和亚黎曼几何的合成里奇曲率下界理论。

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results for some important functional and geometric inequalities, which completely characterize ${\rm CD}(K, \infty)$ condition and non-collapsed ${\rm CD}(K, N)$ condition on Riemannian manifolds with boundary. In particular, using $L^1$-optimal transportation theory, we prove that ${\rm CD}(K, \infty)$ condition implies geodesical convexity.

Functions with bounded variation on a class of Riemannian manifolds with Ricci curvature unbounded from below

After establishing some new global facts (like a measure theoretic structure theorem and approximation results) about complex-valued functions with bounded variation on arbitrary noncompact Riemannian manifolds, we extend results of Miranda/the second author/Paronetto/Preunkert and of Carbonaro/Mauceri on the heat semigroup characterization of the variation of L^1-functions to a class of Riemannian manifolds with possibly unbounded from below Ricci curvature.

Total mean curvatures of Riemannian hypersurfaces

We obtain a comparison formula for integrals of mean curvatures of Riemannian hypersurfaces, via Reilly's identities. As applications we derive several geometric inequalities for a convex hypersurface $Γ$ in a Cartan-Hadamard manifold $M$. In particular we show that the first mean curvature integral of a convex hypersurface $γ$ nested inside $Γ$ cannot exceed that of $Γ$, which leads to a sharp lower bound in dimension $3$ for the total first mean curvature of $Γ$ in terms of the volume it bounds in $M$. This monotonicity property is extended to all mean curvature integrals when $γ$ is paralle…