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Curvature Deformations on Complete Manifolds with Boundary

Tiarlos Cruz, Almir Silva Santos

TL;DR

This work studies conformal deformations of scalar curvature and boundary mean curvature on complete manifolds with boundary, establishing sufficient conditions for obtaining complete metrics with positive scalar curvature and mean convex (or minimal) boundary via a Kazdan-type variational approach that reduces to bounded domains and leverages eigenvalue analyses of the conformal Laplacian with boundary conditions. It develops a Schrödinger-type PDE framework with mixed boundary conditions, introducing variational functionals and positivity criteria that yield positive solutions, enabling the construction of desirable conformal metrics. The paper proves a deformation theorem that, under nonnegative curvature assumptions and nontrivial curvature in either the Ricci tensor or boundary second fundamental form, produces metrics with nonnegative (and often positive) scalar curvature and strictly mean convex boundary, plus a dimension-3 trichotomy describing the global geometric outcomes. It then extends the theory to prescribe negative scalar curvature on noncompact-boundary manifolds via a sub-/super-solution iteration on an exhaustion, obtaining a complete conformal metric with prescribed negative curvature and boundary data, including cases with minimal or positively pinched boundary. Overall, the results broaden deformation techniques for scalar curvature on manifolds with boundary, connecting Kazdan-type ideas, variational eigenvalue methods, and sub-/super-solution frameworks to produce complete metrics with controlled curvature properties.

Abstract

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary. Building upon these results, we explore further deformation scenarios, including those that increase the mean curvature. Finally, we consider the case of deforming complete manifolds with negative scalar curvature on manifolds with noncompact boundary.

Curvature Deformations on Complete Manifolds with Boundary

TL;DR

This work studies conformal deformations of scalar curvature and boundary mean curvature on complete manifolds with boundary, establishing sufficient conditions for obtaining complete metrics with positive scalar curvature and mean convex (or minimal) boundary via a Kazdan-type variational approach that reduces to bounded domains and leverages eigenvalue analyses of the conformal Laplacian with boundary conditions. It develops a Schrödinger-type PDE framework with mixed boundary conditions, introducing variational functionals and positivity criteria that yield positive solutions, enabling the construction of desirable conformal metrics. The paper proves a deformation theorem that, under nonnegative curvature assumptions and nontrivial curvature in either the Ricci tensor or boundary second fundamental form, produces metrics with nonnegative (and often positive) scalar curvature and strictly mean convex boundary, plus a dimension-3 trichotomy describing the global geometric outcomes. It then extends the theory to prescribe negative scalar curvature on noncompact-boundary manifolds via a sub-/super-solution iteration on an exhaustion, obtaining a complete conformal metric with prescribed negative curvature and boundary data, including cases with minimal or positively pinched boundary. Overall, the results broaden deformation techniques for scalar curvature on manifolds with boundary, connecting Kazdan-type ideas, variational eigenvalue methods, and sub-/super-solution frameworks to produce complete metrics with controlled curvature properties.

Abstract

This paper investigates conformal deformations of the scalar curvature and mean curvature on complete Riemannian manifolds with boundary. We establish sufficient conditions for the existence of conformal deformations to complete metrics with positive scalar curvature and mean convex boundary. Building upon these results, we explore further deformation scenarios, including those that increase the mean curvature. Finally, we consider the case of deforming complete manifolds with negative scalar curvature on manifolds with noncompact boundary.
Paper Structure (6 sections, 11 theorems, 65 equations)

This paper contains 6 sections, 11 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Solution to a Geometric PDE
  3. Positive solution of a geometric PDE
  4. Deformations that increases the scalar curvature and the mean curvature
  5. Deformations that increases the mean curvature
  6. Conformal deformation to constant negative scalar Curvature

Key Result

Theorem 1.1

Let $(M^n,g)$ be an $n$-dimensional Riemannian manifold (not necessarily complete or compact), with $n\geq 3$. Assume that $(M^n,g)$ has nonnegative scalar curvature and mean convex boundary. If the Ricci tensor is not identically zero, then there exists a metric $\hat{g}$ on $M$ with the following Moreover, the metric $\hat{g}$ is complete if and only if the original metric $g$ is complete.

Theorems & Definitions (22)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 12 more