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Fractional Sobolev spaces on Riemannian manifolds

Michele Caselli, Enric Florit-Simon, Joaquim Serra

TL;DR

This work develops a canonical theory for the fractional Sobolev energy $H^{s/2}(M)$ on closed Riemannian manifolds with $s\in(0,2)$. It constructs and proves the equivalence of three definitions of the fractional Laplacian—spectral, singular integral, and extension—by exploiting sharp heat-kernel estimates and kernel asymptotics $K_s$, all under precise geometric flatness assumptions. A local monotonicity formula for stationary points of nonlocal energies is established via the Caffarelli–Silvestre extension, yielding insights into $s$-minimal surfaces on manifolds and connecting to a nonlocal version of Yau’s conjecture. The paper also provides detailed estimates for the extension problem and discusses generalizations to noncompact, stochastically complete manifolds, thereby consolidating a robust, canonical framework for nonlocal analysis on manifolds.

Abstract

This article studies the canonical Hilbert energy $H^{s/2}(M)$ on a Riemannian manifold for $s\in(0,2)$, with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type $$ \mathcal E(v)=[v]^2_{H^{s/2}(M)}+\int_M F(v) \, dV \,, \,\,\, F \ge 0 \,, $$ is given, which includes, in particular, the case of nonlocal $s$-minimal surfaces. Finally, we prove some estimates for the Caffarelli-Silvestre extension problem, which are of general interest. This work is motivated by a recent article by the authors, which proves the nonlocal version of a conjecture of Yau.

Fractional Sobolev spaces on Riemannian manifolds

TL;DR

This work develops a canonical theory for the fractional Sobolev energy on closed Riemannian manifolds with . It constructs and proves the equivalence of three definitions of the fractional Laplacian—spectral, singular integral, and extension—by exploiting sharp heat-kernel estimates and kernel asymptotics , all under precise geometric flatness assumptions. A local monotonicity formula for stationary points of nonlocal energies is established via the Caffarelli–Silvestre extension, yielding insights into -minimal surfaces on manifolds and connecting to a nonlocal version of Yau’s conjecture. The paper also provides detailed estimates for the extension problem and discusses generalizations to noncompact, stochastically complete manifolds, thereby consolidating a robust, canonical framework for nonlocal analysis on manifolds.

Abstract

This article studies the canonical Hilbert energy on a Riemannian manifold for , with particular focus on the case of closed manifolds. Several equivalent definitions for this energy and the fractional Laplacian on a manifold are given, and they are shown to be identical up to explicit multiplicative constants. Moreover, the precise behavior of the kernel associated with the singular integral definition of the fractional Laplacian is obtained through an in-depth study of the heat kernel on a Riemannian manifold. Furthermore, a monotonicity formula for stationary points of functionals of the type is given, which includes, in particular, the case of nonlocal -minimal surfaces. Finally, we prove some estimates for the Caffarelli-Silvestre extension problem, which are of general interest. This work is motivated by a recent article by the authors, which proves the nonlocal version of a conjecture of Yau.
Paper Structure (14 sections, 28 theorems, 288 equations)

This paper contains 14 sections, 28 theorems, 288 equations.

Table of Contents

  1. Introduction
  2. Overview of the kernel estimates
  3. The fractional Laplacian on a closed manifold
  4. Spectral and singular integral definitions
  5. Properties of the kernel
  6. Heat kernel estimates
  7. Estimates for the singular kernel
  8. Extension definition
  9. The fractional Sobolev energy
  10. Several definitions and their equivalence
  11. A note on noncompact manifolds
  12. Monotonicity formula for stationary points of semilinear elliptic functionals and -minimal surfaces
  13. The fractional Sobolev energy under inner variations
  14. Estimates for the extension problem

Key Result

Theorem 1.4

Let $(M^n, g)$ be an $n$-dimensional, closed Riemannian manifold. Let $s \in (0,2)$ and where $F$ is any smooth nonnegative function. Let $u:M\to \mathbb{R}$ be stationary for $\mathcal{E}$ under inner variations, meaning that $\mathcal{E}(u)<\infty$ and for any smooth vector field $X$ on $M$ there holds $\frac{d}{dt}|_{t=0} \mathcal{E}(u\circ\psi_X^t)=0$, where $\psi_X^t$ is the flow where $U$

Theorems & Definitions (71)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: Monotonicity formula
  • Definition 2.1: Spectral definition
  • Remark 2.2
  • Definition 2.3: Singular integral definition
  • Remark 2.4
  • Proposition 2.5
  • Remark 2.6
  • ...and 61 more