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Nonlinear Obata type equation on Finsler manifolds without boundary
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Abstract
We show that if there is a nonconstant function
f
∈
C
1,
α
(
M
) satisfying the Obata equation on a Finsler
n
-manifold (
M
,
F
), then (
M
,
F
) is isometric to a Finsler
n
-sphere. In particular, for a Randers metric,
F
is determined analytically by the standard sphere metric and a homothetic field, and the
S
-curvature is constant under the Busemann-Hausdorff volume form. As applications, we prove Lichnerowicz-Obata type rigidity theorem: for a closed Finsler
n
-manifold with the weighted Ricci curvature Ric
N
≥ (
N
− 1)
k
> 0, the first eigenvalue of the Finsler Laplacian is bounded below by
Nk
with equality if and only if (
M
,
F
) is isometric to a Finsler
n
-sphere.
Title: Nonlinear Obata type equation on Finsler manifolds without boundary
Description:
Abstract
We show that if there is a nonconstant function
f
∈
C
1,
α
(
M
) satisfying the Obata equation on a Finsler
n
-manifold (
M
,
F
), then (
M
,
F
) is isometric to a Finsler
n
-sphere.
In particular, for a Randers metric,
F
is determined analytically by the standard sphere metric and a homothetic field, and the
S
-curvature is constant under the Busemann-Hausdorff volume form.
As applications, we prove Lichnerowicz-Obata type rigidity theorem: for a closed Finsler
n
-manifold with the weighted Ricci curvature Ric
N
≥ (
N
− 1)
k
> 0, the first eigenvalue of the Finsler Laplacian is bounded below by
Nk
with equality if and only if (
M
,
F
) is isometric to a Finsler
n
-sphere.
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