Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Calderón's problem for harmonic maps

Sebastián Muñoz-Thon

Abstract

We study a version of Calderón's problem for harmonic maps between Riemannian manifolds. By using the higher linearization method, we first show that the Dirichlet-to-Neumann map determines the metric on the domain up to a natural gauge in three cases: on surfaces, on analytic manifolds, and in conformally transversally anisotropic manifolds on a fixed conformal class with injective ray transform on the transversal manifold. Next, using higher linearizations we obtain integral identities that allows us to show that the metrics on the target have the same jets at one point. In particular, if the target is analytic, the metrics are equal. We also prove an energy rigidity result, in the sense that the Dirichlet energies of harmonic maps determines the Dirichlet-to-Neumann map.

A Calderón's problem for harmonic maps

Abstract

We study a version of Calderón's problem for harmonic maps between Riemannian manifolds. By using the higher linearization method, we first show that the Dirichlet-to-Neumann map determines the metric on the domain up to a natural gauge in three cases: on surfaces, on analytic manifolds, and in conformally transversally anisotropic manifolds on a fixed conformal class with injective ray transform on the transversal manifold. Next, using higher linearizations we obtain integral identities that allows us to show that the metrics on the target have the same jets at one point. In particular, if the target is analytic, the metrics are equal. We also prove an energy rigidity result, in the sense that the Dirichlet energies of harmonic maps determines the Dirichlet-to-Neumann map.

Paper Structure

This paper contains 9 sections, 22 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. High Order Linearization
  3. Integral Identities
  4. Proof of the Main Theorem
  5. Determination of the metric on the domain
  6. Determination of the first derivative of the target's metric
  7. Determination of the second derivative of the target's metric
  8. Determination of the jet of target's metric
  9. Energy Rigidity

Key Result

Theorem 1.1

Let $M$ be a smooth compact manifold with smooth boundary, endowed with the Riemannian metrics $g,\hat{g}$. Let $\Omega \subset \mathbb{R}^{n}$ be a domain with Riemannian metrics $h,\hat{h}$ so that the sectional curvature of $N$ is non-positive, and that $\partial \Omega$ is strictly convex, both where $q \in N$ and $B_{r}(q)$ denotes a ball on $\mathbb{R}^{n}$ with the Euclidean metric. Then w

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 2.1: ES95*p. 59-60
  • Lemma 2.2
  • Lemma 3.1: Integral identity for the second linearization
  • proof
  • Lemma 3.2: Integral identity for the third linearization
  • ...and 31 more