Epsilon regularity under scalar curvature and entropy lower bounds and volume upper bounds

Examples show that Riemannian manifolds with almost-Euclidean lower bounds on scalar curvature and Perelman entropy need not be close to Euclidean space in any metric space sense. Here we show that if one additionally assumes an almost-Euclidean upper bound on volumes of geodesic balls, then unit balls in such a space are Gromov-Hausdorff close, and in fact bi-H\"{o}lder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We prove a compactness and limit space structure theorem under the same assumptions.Comment: Missing reference adde

Minimality and stability properties in Sobolev and isoperimetric inequalities

This thesis addresses the characterization of minimizers in various Sobolev- and isoperimetric-type inequalities and the analysis of the corresponding stability phenomena. We first investigate a family of variational problems which arise in connection to a suitable interpolation between the classical Sobolev and Sobolev trace inequalities. We provide a full characterization of minimizers for each problem, in turn deriving a new family of sharp constrained Sobolev inequalities on the half-space. We then prove novel stability results for the Sobolev inequality and for the anisotropic isoperimetric inequality. Both of these results share the feature of being “strong-form” stability results, in the sense that the deficit in the inequality is shown to control the strongest possible distance to the family of equality cases.Mathematic

A note on the stability of the Cheeger constant of NN-gons

The regular $N$-gon provides the minimal Cheeger constant in the class of all $N$-gons with fixed volume. This result is due to a work of Bucur and Fragal\`a in 2014. In this note, we address the stability of their result in terms of the $L^1$ distance between sets. Furthermore, we provide a stability inequality in terms of the Hausdorff distance between the boundaries of sets in the class of polygons having uniformly bounded diameter. Finally, we show that our results are sharp, both in the exponent of decay and in the notion of distance between sets.Comment: 5 pages, 2 figure

Linear Stability Implies Nonlinear Stability for Faber-Krahn Type Inequalities

For a domain $\Omega \subset \mathbb{R}^n$ and a small number $\frak{T} > 0$, let $$\mathcal{E}_0(\Omega) = \lambda_1(\Omega) + {\frak{T}} {\text{tor}}(\Omega) = \inf_{u, w \in H^1_0(\Omega)\setminus \{0\}} \frac{\int |\nabla u|^2}{\int u^2} + {\frak{T}} \int \frac{1}{2} |\nabla w|^2 - w$$ be a modification of the first Dirichlet eigenvalue of $\Omega$. It is well-known that over all $\Omega$ with a given volume, the only sets attaining the infimum of $\mathcal{E}_0$ are balls $B_R$; this is the Faber-Krahn inequality. The main result of this paper is that, if for all $\Omega$ with the same volume and barycenter as $B_R$ and whose boundaries are parametrized as small $C^2$ normal graphs over $\partial B_R$ with bounded $C^2$ norm, $$\int |u_{\Omega} - u_{B_R}|^2 + |\Omega \triangle B_R|^2 \leq C [\mathcal{E}_0(\Omega) - \mathcal{E}_0(B_R)]$$ (i.e. the Faber-Krahn inequality is linearly stable), then the same is true for any $\Omega$ with the same volume and barycenter as $B_R$ without any smoothness assumptions (i.e. it is nonlinearly stable). Here $u_{\Omega}$ stands for an $L^2$-normalized first Dirichlet eigenfunction of $\Omega$. Related results are shown for Riemannian manifolds. The proof is based on a detailed analysis of some critical perturbations of Bernoulli-type free boundary problems. The topic of when linear stability is valid, as well as some applications, are considered in a companion paper.Comment: 72 pages, comments welcome

Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit

We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems