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A universal Bochner formula for scalar curvature

Sven Hirsch

TL;DR

The paper develops a universal Bochner framework that unifies minimal slicings, spinors, and level-set methods for encoding scalar curvature. By introducing a system of vector fields $Z_m$ derived from functions $u_m$ and imposing appropriate divergence-free or normalization conditions, it derives a pointwise inequality linking Laplacians, curvature, and gradient terms that specializes to known stability inequalities, a Schrödinger–Lichnerowicz-type bound, and a higher-dimensional Stern identity. These results recover classical nonexistence statements for metrics of positive scalar curvature on certain manifolds, produce new Dirac-current constructions in dimension three, and open avenues toward intermediate curvature theories and potential applications to problems like the Geroch conjecture and positive mass theorems. Overall, the work provides a concrete dictionary connecting three analytic approaches to scalar curvature and suggests novel paths to address long-standing geometric questions.

Abstract

We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schrödinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity.

A universal Bochner formula for scalar curvature

TL;DR

The paper develops a universal Bochner framework that unifies minimal slicings, spinors, and level-set methods for encoding scalar curvature. By introducing a system of vector fields derived from functions and imposing appropriate divergence-free or normalization conditions, it derives a pointwise inequality linking Laplacians, curvature, and gradient terms that specializes to known stability inequalities, a Schrödinger–Lichnerowicz-type bound, and a higher-dimensional Stern identity. These results recover classical nonexistence statements for metrics of positive scalar curvature on certain manifolds, produce new Dirac-current constructions in dimension three, and open avenues toward intermediate curvature theories and potential applications to problems like the Geroch conjecture and positive mass theorems. Overall, the work provides a concrete dictionary connecting three analytic approaches to scalar curvature and suggests novel paths to address long-standing geometric questions.

Abstract

We introduce a universal Bochner formula for scalar curvature that contains, as special cases, the stability inequality for minimal slicings, a Schrödinger-Lichnerowicz-type formula, and a higher-dimensional version of Stern's level-set identity.
Paper Structure (6 sections, 15 theorems, 104 equations)

This paper contains 6 sections, 15 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Stable minimal slicings
  3. Spinors and the Lichnerowicz formula
  4. Level sets and intermediate curvature
  5. Proof of the Bochner formula
  6. Minimal slicings, spinors, and level-sets

Key Result

Theorem 1.1

Let $(M^n,g)$ be a Riemannian manifold, and let $u_0,\dots,u_{n-2}$ be $C^3$ functions satisfying $du_0\wedge\cdots\wedge du_{n-2}\neq 0$. Define vector fields $Z_0,\dots,Z_{n-1}$ iteratively by Suppose that at a point $p\in M$ we have Then at $p$,

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 21 more