# 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 ${\omega}_{1},{\omega}_{2}\in {\mathbb{R}}^{m}$:$$\left(\right)open="("\; close=")">\mathcal{A}(x,{\omega}_{2})-\mathcal{A}(x,{\omega}_{1})$$

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

**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

## References

- Choquet-Bruhat, Y. Beginnings of the Cauchy problem for Einstein’s field equations. In Surveys in Differential Geometry 2015. One Hundred Years of General Relativity; Surveys in Differential Geometry; International Press: Boston, MA, USA, 2015; Volume 20, pp. 1–16. [Google Scholar]
- DeTurck, D.M.; Kazdan, J.L. Some regularity theorems in Riemannian geometry. Annales Scientifiques de L’École Normale Supérieure
**1981**, 14, 249–260. [Google Scholar] [CrossRef] - Müller zum Hagen, H. On the analyticity of static vacuum solutions of Einstein’s equations. Proc. Camb. Philos. Soc.
**1970**, 67, 415–421. [Google Scholar] [CrossRef] - Bers, L.; John, F.; Schechter, M. Partial Differential Equations; Lectures in Applied Mathematics; Interscience Publishers John Wiley & Sons, Inc.: New York, NY, USA; London, UK; Sydney, Australia, 1964; Volume III. [Google Scholar]
- Liimatainen, T.; Salo, M. n-harmonic coordinates and the regularity of conformal mappings. Math. Res. Lett.
**2014**, 21, 341–361. [Google Scholar] [CrossRef] - Taylor, M. Existence and regularity of isometries. Trans. Am. Math. Soc.
**2006**, 358, 2415–2423. [Google Scholar] [CrossRef] - Douglis, A.; Nirenberg, L. Interior estimates for elliptic systems of partial differential equations. Commun. Pure Appl. Math.
**1955**, 8, 503–538. [Google Scholar] [CrossRef] - Morrey, C.B., Jr. On the analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations. I. Analyticity in the interior. Am. J. Math.
**1958**, 80, 198–218. [Google Scholar] [CrossRef] - Shen, Z. Landsberg curvature. In A Sampler of Riemann-Finsler Geometry; Mathematical Sciences Research Institute Publications: Cambridge, UK; Cambridge University Press, 2004; Volume 50, pp. 303–355. [Google Scholar]
- Taylor, M.E. Partial Differential Equations I. Basic Theory, 2nd ed.; Applied Mathematical Sciences; Springer: New York, NY, USA, 2011; Volume 115. [Google Scholar]
- Ge, Y.; Shen, Z. Eigenvalues and eigenfunctions of metric measure manifolds. Proc. Lond. Math. Soc. Third Ser.
**2001**, 82, 725–746. [Google Scholar] [CrossRef] - Ohta, S.I.; Sturm, K.T. Heat flow on Finsler manifolds. Commun. Pure Appl. Math.
**2009**, 62, 1386–1433. [Google Scholar] [CrossRef] [Green Version] - Besse, A.L. Einstein Manifolds; Reprint of the 1987 Edition; Classics in Mathematics; Springer: Berlin, Germany, 2008. [Google Scholar]
- Taylor, M.E. Tools for PDE; Pseudodifferential Operators, Paradifferential Operators, Ed.; Mathematical Surveys and Monographs; American Mathematical Society: Providence, RI, USA, 2000; Volume 81. [Google Scholar]
- Deng, S.; Hou, Z. The group of isometries of a Finsler space. Pac. J. Math.
**2002**, 207, 149–155. [Google Scholar] [CrossRef] - Aradi, B.; Kertész, D.C. Isometries, submetries and distance coordinates on Finsler manifolds. Acta Math. Hung.
**2014**, 143, 337–350. [Google Scholar] [CrossRef] - Matveev, V.S.; Troyanov, M. The Myers-Steenrod theorem for Finsler manifolds of low regularity. Proc. Am. Math. Soc.
**2017**, 145, 2699–2712. [Google Scholar] [CrossRef] [Green Version] - Shen, Z. Lectures on Finsler Geometry; World Scientific Publishing Co.: Singapore, 2001. [Google Scholar]
- Szilasi, J.; Lovas, R.L.; Kertész, D.C. Several ways to a Berwald manifold—And some steps beyond. Extr. Math.
**2011**, 26, 89–130. [Google Scholar] - Bao, D.; Chern, S.S.; Shen, Z. An Introduction to Riemann-Finsler Geometry; Graduate Texts in Mathematics; Springer: New York, NY, USA, 2000. [Google Scholar]
- Szabó, Z.I. Positive definite Berwald spaces. Structure theorems on Berwald spaces. Tensor Soc. Tensor New Ser.
**1981**, 35, 25–39. [Google Scholar] - Crampin, M. On the construction of Riemannian metrics for Berwald spaces by averaging. Houst. J. Math.
**2014**, 40, 737–750. [Google Scholar] - Vincze, C. A new proof of Szabó’s theorem on the Riemann-metrizability of Berwald manifolds. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis
**2005**, 21, 199–204. [Google Scholar] - Matveev, V.S. Riemannian metrics having common geodesics with Berwald metrics. Publ. Math. Debr.
**2009**, 74, 405–416. [Google Scholar] - Matveev, V.S.; Rademacher, H.B.; Troyanov, M.; Zeghib, A. Finsler conformal Lichnerowicz-Obata conjecture. Annales De L’Institut Fourier
**2009**, 59, 937–949. [Google Scholar] [CrossRef] - Matveev, V.S.; Troyanov, M. The Binet-Legendre metric in Finsler geometry. Geom. Topol.
**2012**, 16, 2135–2170. [Google Scholar] [CrossRef] - Bao, D.; Lackey, B. A geometric inequality and a Weitzenböck formula for Finsler surfaces. In The Theory of Finslerian Laplacians and Applications; Mathematics and Its Applications; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1998; Volume 459, pp. 245–275. [Google Scholar]
- Dragomir, S.; Larato, B. Harmonic functions on Finsler spaces. Univ. Istanbul. Fac. Sci. J. Math.
**1989**, 48, 67–76. [Google Scholar] - Caponio, E.; Javaloyes, M.A.; Masiello, A. Finsler geodesics in the presence of a convex function and their applications. J. Phys. A
**2010**, 43, 135207. [Google Scholar] [CrossRef]

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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