Colding established a sharp gradient estimate for the Green function on manifolds with nonnegative Ricci curvature, which was extended to positive Ricci curvature by Manea recently. We prove almost rigidity of such gradient estimates for positive Ricci curvature. We show that the average of the gradient deficit, namely $\fint (1-|\nabla b|^2)\,dV$, controls the volume deficit in a quantitative manner; here $b$ is the distance-like function defined using the Green function. This implies a quantitative almost rigidity theorem for manifolds with $\mathrm{Ric}\ge(n-1)g$ and a gap theorem for Einstein manifolds. We also prove a rigidity theorem for closed 4-dimensional Einstein manifolds by finding a new monotonicity formula.
We establish a weighted extension of the Bishop-Gromov volume comparison theorem for complete Riemannian manifolds with Ricci curvature bounded below. Given a point $p$ and positive radial weights $f$ and $h$, we consider the weighted volume of a geodesic ball and a corresponding model weighted volume in the simply con...
We prove that the weighted Laplacian, or equivalently its conjugate Schr\"odinger operator, on every complete gradient Ricci shrinker satisfies the classical Weyl law. The main difficulty is that uniform bounded geometry is not known for general Ricci shrinkers. To overcome this, we establish a large-scale regularity p...
We prove Petrunin's smoothing conjecture in all dimensions: every compact Euclidean polyhedral space without boundary and with nonnegative Alexandrov curvature is a Gromov-Hausdorff limit of smooth Riemannian orbifolds with geometrically nonnegative curvature. More strongly, the approximating metrics are positive-time...
Let $n\geq3$ and $0\leq\tau\leq2$. We prove that every smooth closed connected immersed hypersurface in hyperbolic space whose induced metric has positive $\tau$-bi-Ricci curvature admits a mean curvature flow with surgery which has only finitely many surgery times and terminates. In the range compatible with cylindric...
Let G(p, .) be the normalized minimal positive Green function on a complete nonparabolic Riemannian manifold with nonnegative Ricci curvature, and set b = G^(1/(2-n)). Colding established the following rigidity result in dimension three: if |nabla b|^2 admits a C^2 regularity across the pole, then the manifold must be...
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved...
Shi-Bing Chen, M. Ghomi, Peng Wang· 3 citations· ⚡1
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