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Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature

Alexandru Kristály, Benling Li, Wei Zhao

Abstract

The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and $S$-curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive $S$-curvature, cf. Huang, Kristály and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the $S$-curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the $S$-curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the $S$-curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the $S$-curvature plays again a decisive role.

Failure of famous functional inequalities on Finsler manifolds: the influence of $S$-curvature

Abstract

The validity of functional inequalities on Finsler metric measure manifolds is based on three non-Riemannian quantities, namely, the reversibility, flag curvature and -curvature induced by the measure. Under mild assumptions on the reversibility and flag curvature, it turned out that famous functional inequalities -- as Hardy inequality, Heisenberg--Pauli--Weyl uncertainty principle and Caffarelli--Kohn--Nirenberg inequality -- usually hold on forward complete Finsler manifolds with non-positive -curvature, cf. Huang, Kristály and Zhao [Trans. Amer. Math. Soc., 2020]. In this paper however we prove that -- under similar assumptions on the reversibility and flag curvature as before -- the aforementioned functional inequalities fail whenever the -curvature is positive. Accordingly, our results clearly reveal the deep dependence of functional inequalities on the -curvature. As a consequence of these results, we establish analytic aspects of Finsler manifolds, e.g., if the flag curvature is non-positive, the Ricci curvature is bounded from below and the -curvature is positive, then the reversibility turns out to be infinite. Further topological properties and examples are presented on general Funk metric spaces, where the -curvature plays again a decisive role.
Paper Structure (17 sections, 29 theorems, 230 equations)

This paper contains 17 sections, 29 theorems, 230 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Elements from Finsler geometry
  4. Finsler manifolds
  5. Measure and polar coordinate system
  6. Sobolev spaces over Finsler manifolds and approximation
  7. Failure of functional inequalities: proof of Theorem \ref{['mainThm1']}
  8. Vanishing first eigenvalue of $p$-Laplacian
  9. Failure of Hardy inequality
  10. Failure of Heisenberg--Pauli--Weyl uncertainty principle
  11. Failure of Caffarelli--Kohn--Nirenberg interpolation inequality
  12. Influence of $S$-curvature on geometric aspects: proof of Theorems \ref{['reversibinfite']} &\ref{['Spositivenoncom']}
  13. Infinite reversibility
  14. Topological properties of projectively flat manifolds
  15. Examples & Counterexamples: Funk metric spaces
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $n\geq 2$ be an integer, $(M,F,\mathop{\mathrm{\mathfrak{m}}}\nolimits)$ be an $n$-dimensional forward complete $\mathop{\mathrm{\mathsf{FMMM}}}\nolimits$, and $r=d_F(o,\cdot)$ for some $o\in M$. Suppose that where $k,h$ are two constants with $h> k\geq 0$. Then the following statements hold$:$

Theorems & Definitions (66)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Definition 3.1
  • Remark 3.1
  • Proposition 3.1
  • ...and 56 more