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Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function

Abstract We prove a matrix Li–Yau–Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate is an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds. It is also the complex counterpart of the matrix estimate on Riemannian manifolds obtained previously by the author.

We give a natural way to identify between two scales, potentially arbitrarily far apart, in a non-compact Ricci-flat manifold with Euclidean volume growth when a tangent cone at infinity has smooth cross section. The identification map is given as the gradient flow of a solution to an elliptic equation.

In this paper we prove a matrix Li-Yau-Hamilton inequality for the Green function on complete Kähler manifolds with nonnegative holomorphic bisectional curvature. This estimate can be seen as an elliptic analogue of the matrix estimate of Cao and Ni for the heat equation on Kähler manifolds, or the complex analogue of the estimate for Riemannian manifolds obtained previously by the author.

Abstract Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler–Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold wi

Gromov and Sormani conjectured that sequences of compact Riemannian manifolds with nonnegative scalar curvature and area of minimal surfaces bounded below should have subsequences which converge in the intrinsic flat sense to limit spaces which have nonnegative generalized scalar curvature and Euclidean tangent cones almost everywhere. In this paper we prove this conjecture for sequences of rotationally symmetric warped product manifolds. We show that the limit spaces have $H^1$ warping function

We prove three monotonicity formulas (which imply rigidity theorems when equality holds) for nonparabolic manifolds satisfying a matrix Harnack estimate which we call Property $(H G)_C$ (that is, the Hessian of the $2 /(2-n)$ power of the renormalized Green function is uniformly bounded by $C g$ ). This Property $(H G)_C$ is satisfied on manifolds that meet certain conditions including bounds on the sectional curvature and covariant derivative of the Ricci curvature, as shown in the author's pre

In \cite{Colding}, Colding proved monotonicity formulas for the Green function on manifolds with nonnegative Ricci curvature. Inspired by the sharp estimates relating the pinching of monotone quantities to the splitting function in \cite{cjn}, in this paper we investigate quantitative control obtained from pinching of Colding's monotone functionals. From the Green functions with poles at $(k+1)$-many independent points, $k$-splitting functions are constructed with regularity quantitatively contr

We prove three related quantitative results for the relative isoperimetric problem outside a convex body $Ω$ in the plane: (1) Łojasiewicz estimates and quantitative rigidity for critical points, (2) rates of convergence for the gradient flow, and (3) quantitative stability for minimizers. These results come with explicit constants and optimal exponents/rates, and hold whenever a simple two-dimensional auxiliary variational problem for circular arcs outside of $Ω$ is nondegenerate. The proofs ar