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$L^2$-estimates on flat vector bundles and Prékopa's theorem

Gang Huang, Weiwen Jiang, Xiangsen Qin

Abstract

In this paper, we will construct Hörmander's $L^2$-estimate of the operator $d$ on a flat vector bundle over a $p$-convex Riemannian manifold and discuss some geometric applications of it. In particular, we will generalize the classical Prékopa's theorem in convex analysis.

$L^2$-estimates on flat vector bundles and Prékopa's theorem

Abstract

In this paper, we will construct Hörmander's -estimate of the operator on a flat vector bundle over a -convex Riemannian manifold and discuss some geometric applications of it. In particular, we will generalize the classical Prékopa's theorem in convex analysis.
Paper Structure (6 sections, 17 theorems, 81 equations)

This paper contains 6 sections, 17 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. The proof of Theorem \ref{['thm:flat vector bundle']}
  4. The proof of Theorem \ref{['thm:improved $L^2$-estimates']}
  5. The proof of Theorem \ref{['thm:L^2 inverse']}
  6. The proofs of Theorem \ref{['thm:limit metric']} and \ref{['thm:prekopa theorem']}

Key Result

Theorem 1.1

Let $M$ be an $n$-dimensional $p$-convex Riemannian manifold without boundary for some $p\in\{1,\cdots,n\}$, $(E,h)$ be a flat vector bundle over $M$. Suppose $\mathfrak{Ric}_p+\Theta^{(E,h)}\geq 0,$ then for any $d$-closed $f\in L^2_{\mathop{\mathrm{loc}}\nolimits}(M,\Lambda^pT^*M\otimes E)$ satisf there exists $u\in L^2(M,\Lambda^pT^*M\otimes E)$ such that Moreover, if $f$ is smooth, then $u$ c

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 17 more