Density and spectrum of minimal submanifolds in space forms
Let φ : M m → N n be a minimal, proper immersion in an ambient space suitably close to a space form N k n of curvature - k ≤ 0 . In this paper, we are interested in the relation between the density function Θ ( r ) of M and the spectrum of its Laplace–Beltrami operator. In particular, we prove that...
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Published in | Mathematische annalen Vol. 366; no. 3-4; pp. 1035 - 1066 |
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Main Authors | , , , |
Format | Journal Article |
Language | English |
Published |
Berlin/Heidelberg
Springer Berlin Heidelberg
01.12.2016
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | Let
φ
:
M
m
→
N
n
be a minimal, proper immersion in an ambient space suitably close to a space form
N
k
n
of curvature
-
k
≤
0
. In this paper, we are interested in the relation between the density function
Θ
(
r
)
of
M
and the spectrum of its Laplace–Beltrami operator. In particular, we prove that if
Θ
(
r
)
has subexponential growth (when
k
<
0
) or sub-polynomial growth (
k
=
0
) along a sequence, then the spectrum of
M
m
is the same as that of the space form
N
k
m
. Notably, the result applies to Anderson’s (smooth) solutions of Plateau’s problem at infinity on the hyperbolic space, independently of their boundary regularity. We also give a simple condition on the second fundamental form that ensures
M
to have finite density. In particular, we show that minimal submanifolds with finite total curvature in the hyperbolic space also have finite density. |
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ISSN: | 0025-5831 1432-1807 |
DOI: | 10.1007/s00208-016-1360-y |