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Concavity for elliptic and parabolic equations in locally symmetric spaces with nonnegative curvature

Shrey Aryan, Michael B. Law

Abstract

We establish a concavity principle for solutions to elliptic and parabolic equations on locally symmetric spaces with nonnegative sectional curvature, extending the results of Langford and Scheuer. To the best of our knowledge, this is the first general concavity principle established on spaces with non-constant sectional curvature.

Concavity for elliptic and parabolic equations in locally symmetric spaces with nonnegative curvature

Abstract

We establish a concavity principle for solutions to elliptic and parabolic equations on locally symmetric spaces with nonnegative sectional curvature, extending the results of Langford and Scheuer. To the best of our knowledge, this is the first general concavity principle established on spaces with non-constant sectional curvature.
Paper Structure (13 sections, 12 theorems, 64 equations)

This paper contains 13 sections, 12 theorems, 64 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Proof sketch
  4. Organization
  5. Jacobi fields on locally symmetric spaces
  6. Setup and frame fixing
  7. Spatial derivatives of Jacobi fields
  8. Concavity principles
  9. Proofs of main theorems
  10. Boundary conditions and examples
  11. Relaxing structural assumptions
  12. Boundary condition
  13. Examples

Key Result

Theorem 1.1

Let $\mathcal{M}$ be a locally symmetric space of dimension $n \geq 1$ with sectional curvatures lying in the interval $[0,A]$. Let $\Omega \subset \mathcal{M}$ be a domain with geodesically convex closure and diameter strictly less than $\frac{\pi}{\sqrt{A}}$. Let $\Gamma \subset \mathbb{R}^n$ be a Suppose the function $f: [0,\infty) \times \mathcal{D}_{\bar{\Gamma}}$ has the following properties

Theorems & Definitions (24)

  • Theorem 1.1: Elliptic concavity principle
  • Theorem 1.2: Parabolic concavity principle
  • Remark 1.3: Isotropic functions and the domain of $f$
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Lemma 2.3: langford2021concavity*Lemma 3.1
  • Lemma 2.4
  • proof
  • ...and 14 more