Regularity of conformal metrics with large first eigenvalue

Henrik Matthiesen

doi:10.5802/afst.1523

Henrik Matthiesen ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 1079-1094.

Abstract
Abstract

We establish a regularity result for conformal metrics with unit volume, $L^{p}$ scalar curvature bounds for $p > n / 2$ and first eigenvalue of $Δ$ bounded from below by a constant $B > Λ_{1} (S^{n}, [g_{s t .}]) .$

On démontre un résultat sur la régularité de métriques conformes de volumes unitaires avec une borne supérieure sur la norme $L^{p}$ de la courbure scalaire pour $p > n / 2$ , et une borne inférieure sur la première valeur propre de $Δ$ par une constante $B > Λ_{1} (S^{n}, [g_{s t .}]) .$

Published online: 2016-11-13

MR Zbl

DOI: 10.5802/afst.1523

Author's affiliations:

Henrik Matthiesen ¹

¹ Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn

@article{AFST_2016_6_25_5_1079_0,
     author = {Henrik Matthiesen},
     title = {Regularity of conformal metrics with large first eigenvalue},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {1079--1094},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 25},
     number = {5},
     year = {2016},
     doi = {10.5802/afst.1523},
     mrnumber = {3582121},
     zbl = {1373.53052},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1523/}
}

TY  - JOUR
AU  - Henrik Matthiesen
TI  - Regularity of conformal metrics with large first eigenvalue
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2016
SP  - 1079
EP  - 1094
VL  - 25
IS  - 5
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1523/
DO  - 10.5802/afst.1523
LA  - en
ID  - AFST_2016_6_25_5_1079_0
ER  -

%0 Journal Article
%A Henrik Matthiesen
%T Regularity of conformal metrics with large first eigenvalue
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2016
%P 1079-1094
%V 25
%N 5
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1523/
%R 10.5802/afst.1523
%G en
%F AFST_2016_6_25_5_1079_0

Henrik Matthiesen. Regularity of conformal metrics with large first eigenvalue. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 25 (2016) no. 5, pp. 1079-1094. doi : 10.5802/afst.1523. https://afst.centre-mersenne.org/articles/10.5802/afst.1523/

References
Cited by

[1] Brooks (R.), Perry (P.), Yang (P.).— Isospectral sets of conformally equivalent metrics. Duke Math. J. 58, no.1, p. 131-150 (1989). | DOI | Zbl

[2] Chang (S.-Y. A.), Gursky (M.), Wolff (T).— Lack of compactness in conformal metrics with $L^{d / 2}$ curvature. J. Geom. Anal. 4, no.2, p. 143-153 (1994). | DOI | Zbl

[3] Chang (S.-Y. A.), Yang (P. C.-P.).— Compactness of isospectral conformal metrics on $S^{3}$ . Comment. Math. Helv. 64, no.3, p. 363-374 (1989). | DOI | Zbl

[4] Chang (S.-Y. A.), Yang (P. C.-P.).— Isospectral conformal metrics on $3$ -manifolds. J. Amer. Math. Soc. 3, no.1, p. 117-145 (1990). | DOI

[5] Colbois (B.), El Soufi (A.).— Extremal eigenvalues of the Laplacian in a conformal class of metrics: the ‘conformal spectrum’. Ann. Global Anal. Geom. 24, no.4 p. 337-349 (2003). | DOI | Zbl

[6] EL Soufi (A.), Ilias (S.).— Immersions minimales, première valeur propre du laplacien et volume conforme. Math. Ann. 275, no.2, p. 257-267 (1986). | DOI | Zbl

[7] Gursky (M.).— Compactness of conformal metrics with integral bounds on curvature. Duke Math. J. 72, no.2, p. 339-367 (1993). | DOI | Zbl

[8] Hersch (J.).— Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270, A1645-A1648 (1993). | Zbl

[9] Kokarev (G.).— Variational aspects of Laplace eigenvalues on Riemannian surfaces. Adv. Math. 258, p. 191-239 (2014). | DOI | Zbl

[10] Korevaar (N.).— Upper bounds for eigenvalues of conformal metrics. J. Differ. Geom. 37, no.1, p. 73-93 (1993). | DOI | Zbl

[11] Li (P.), Yau (S. T.).— A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math 69, no.2, p. 269-291 (1982). | DOI | Zbl

[12] Morrey Jr. (C. B.).— Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften Band 130, ix+506 pp. (1966). | DOI

[13] Nadirashvili (N.).— Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal. 6, no.5, p. 877-897 (1996). | DOI | Zbl

[14] Petrides (R.).— On a rigidity result for the first conformal eigenvalue of the Laplacian. J. Spectr. Theory 5, no.1 p. 227-234 (2015). | DOI | Zbl

[15] Petrides (R.).— Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces. Geom. Funct. Anal. 4, no.4 p. 1336-1376 (2014). | DOI | Zbl

[16] Trudinger (N. S.).— Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Scuola Norm. Sup. Pisa (3) 22, p. 265-274 (1968). | Zbl

Cited by Sources: