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Unique continuation estimates on manifolds with Ricci curvature bounded below

Christian Rose, Martin Tautenhahn

Abstract

We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schrödinger operators on Riemannian manifolds with Ricci curvature bounded below. The upper bound for the energy range and the constant appearing in the estimate are given in terms of the lower bound of the Ricci curvature and the parameters of the relatively dense set.

Unique continuation estimates on manifolds with Ricci curvature bounded below

Abstract

We prove quantitative unique continuation estimates for relatively dense sets and spectral subspaces associated to small energies of Schrödinger operators on Riemannian manifolds with Ricci curvature bounded below. The upper bound for the energy range and the constant appearing in the estimate are given in terms of the lower bound of the Ricci curvature and the parameters of the relatively dense set.
Paper Structure (6 sections, 16 theorems, 109 equations)

This paper contains 6 sections, 16 theorems, 109 equations.

Table of Contents

  1. Introduction and main result
  2. Norm estimates for the semigroup at large coupling and an abstract quantitative unique continuation estimate
  3. Spectral estimates for Laplacians on domains with relatively dense complement
  4. Consequences of Ricci curvature lower bounds and the main result
  5. Quantitative unique continuation estimates and Ricci curvature
  6. Acknowledgement

Key Result

Theorem 1.1

Let $K\in\mathbb{R}$, $n\in\mathbb{N}$, $n\geq 3$, and $0<\rho<R$. There are $\kappa=\kappa(K,R,\rho,n)>0$ and $E_0=E_0(K,R,\rho,n,a,b) \in \mathbb{R}$ such that for any complete Riemannian manifold $M$ of dimension $n$ with $\mathop{\mathrm{Ric}}\nolimits\geq K$, any proper $(R,\rho)$-relatively de where the inequality holds in quadratic form sense.

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof : Proof of Lemma \ref{['lemma:hitrun']}
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • proof
  • Theorem 2.5
  • ...and 20 more