# Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

## 2. The Nonlinear Finsler Laplacian

- $F(x,\lambda v)=\lambda F(x,v)$ for all $\lambda >0$ and $v\in TM$, $F(x,v)=0$ if and only if $v=0$;
- the bilinear symmetric form on ${T}_{x}M$, depending on $(x,v)\in TM\backslash 0$,$$g(x,v)[{w}_{1},{w}_{2}]:=\phantom{\rule{-0.166667em}{0ex}}\frac{1}{2}\frac{{\partial}^{2}}{\partial s\partial t}{F}^{2}(x,v+s{w}_{1}+t{w}_{2}){|}_{(s,t)=(0,0)}$$

- for all $(x,\omega )\in U\times {\mathbb{R}}^{m}\backslash \left\{0\right\}$:$$\parallel \mathcal{A}(x,\omega )\parallel +\parallel {\partial}_{x}\mathcal{A}(x,\omega )\parallel +\parallel \omega \parallel \phantom{\rule{0.166667em}{0ex}}\parallel {\partial}_{\omega}\mathcal{A}(x,\omega )\parallel \le C\parallel \omega \parallel ;$$
- for all $(x,\omega )\in U\times ({\mathbb{R}}^{m}\backslash \left\{0\right\})$ and all $h\in {\mathbb{R}}^{m}$:$${\partial}_{\omega}\mathcal{A}(x,\omega )[h,h]\ge \frac{1}{C}{\parallel h\parallel}^{2};$$
- for all $x\in U$ and ${\omega}_{1},{\omega}_{2}\in {\mathbb{R}}^{m}$:$$\left(\mathcal{A}(x,{\omega}_{2})-\mathcal{A}(x,{\omega}_{1})\right)({\omega}_{2}-{\omega}_{1})\ge \frac{1}{C}{\parallel {\omega}_{2}-{\omega}_{1}\parallel}^{2};$$

**Proposition**

**1.**

**Remark**

**1.**

**Proposition**

**2.**

## 3. Harmonic Coordinates in Finsler Manifolds

**Proof**

**of**

**Theorem**

**1.**

**Lemma**

**1.**

- (a)
- ${F}_{1}={\mathcal{I}}^{*}\left({F}_{2}\right)$, (i.e., ${F}_{1}(x,v)={F}_{2}(\mathcal{I}\left(x\right),d\mathcal{I}\left(x\right)\left[v\right])$);
- (b)
- ${\ell}_{1}^{-1}(x,\omega )=\left(x,d{\mathcal{I}}^{-1}\left(\mathcal{I}\left(x\right)\right)\left[{\mathcal{J}}_{2,\mathcal{I}\left(x\right)}^{-1}(\omega \circ d{\mathcal{I}}^{-1})\right]\right)$;
- (c)
- ${F}_{1}^{*}={\mathcal{I}}^{*}\left({F}_{2}^{*}\right)$, (i.e., ${F}_{1}^{*}(x,\omega )={F}_{2}^{*}\left(\mathcal{I}\left(x\right),\omega \circ d{\mathcal{I}}^{-1}\right)$).

**Proof.**

**Proposition**

**3.**

**Proof.**

**Remark**

**2.**

## 4. Regularity Results for Berwald Metrics

**Theorem**

**2**

**.**Let h be a ${C}^{2}$ Riemannian metric, and $\mathrm{Ric}\left(h\right)$ be its Ricci tensor. If, in harmonic coordinates of h, $\mathrm{Ric}\left(h\right)$ is of class ${C}^{k,\alpha}$, for $k\ge 0$, (respectively, ${C}^{\infty}$ and ${C}^{\omega}$), then, in these coordinates, h is of class ${C}^{k+2,\alpha}$ (respectively, ${C}^{\infty}$ and ${C}^{\omega}$).

**Proposition**

**4.**

**Proof.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Caponio, E.; Masiello, A.
Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics. *Axioms* **2019**, *8*, 83.
https://doi.org/10.3390/axioms8030083

**AMA Style**

Caponio E, Masiello A.
Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics. *Axioms*. 2019; 8(3):83.
https://doi.org/10.3390/axioms8030083

**Chicago/Turabian Style**

Caponio, Erasmo, and Antonio Masiello.
2019. "Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics" *Axioms* 8, no. 3: 83.
https://doi.org/10.3390/axioms8030083