A second derivative formula of the Liouville entropy at
spaces of constant negative curvature

In this paper we study a new functional on the space of metrics with negative curvature on a compact manifold. It is a linear combination of Liouville entropy and total scalar curvature. Locally symmetric spaces are critical points of this functional. We provide an explicit formula for its second derivative at metrics of constant negative curvature. In particular, this shows that a metric of constant curvature is a local maximum.

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The Minakshisundaram-Pleijel Coefficients for the Vector Valued Heat Kernel on Compact Locally Symmetric Spaces of Negative Curvature

Transactions of the American Mathematical Society ◽

10.2307/1999874 ◽

1980 ◽

Vol 260 (1) ◽

pp. 1 ◽

Cited By ~ 2

Author(s):

Roberto J. Miatello

Keyword(s):

Heat Kernel ◽

Negative Curvature ◽

Symmetric Spaces ◽

Locally Symmetric Spaces ◽

Vector Valued

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A more conclusive and more inclusive second derivative test

The Mathematical Gazette ◽

10.1017/mag.2020.47 ◽

2020 ◽

Vol 104 (560) ◽

pp. 247-254

Author(s):

Ronald Skurnick ◽

Christopher Roethel

Keyword(s):

Critical Point ◽

Critical Points ◽

Local Minimum ◽

Local Maximum ◽

Number Line ◽

Second Derivative ◽

First Derivative ◽

Line Test ◽

First Derivative Test ◽

Derivative Test

Given a differentiable function f with argument x, its critical points are those values of x, if any, in its domain for which either f′ (x) = 0 or f′ (x) is undefined. The first derivative test is a number line test that tells us, definitively, whether a given critical point, x = c, of f(x) is a local maximum, a local minimum, or neither. The second derivative test is not a number line test, but can also be applied to classify the critical points of f(x). Unfortunately, the second derivative test is, under certain conditions, inconclusive.

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On locally symmetric spaces of non-negative curvature and certain other locally homogeneous spaces

Commentarii Mathematici Helvetici ◽

10.1007/bf02566977 ◽

1962 ◽

Vol 37 (1) ◽

pp. 266-295 ◽

Cited By ~ 9

Author(s):

Joseph A. Wolf

Keyword(s):

Negative Curvature ◽

Symmetric Spaces ◽

Homogeneous Spaces ◽

Locally Symmetric Spaces ◽

Locally Homogeneous

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Entropy-Rigidity of Locally Symmetric Spaces of Negative Curvature

Annals of Mathematics ◽

10.2307/1971507 ◽

1990 ◽

Vol 131 (1) ◽

pp. 35 ◽

Cited By ~ 14

Author(s):

Ursula Hamenstadt

Keyword(s):

Negative Curvature ◽

Symmetric Spaces ◽

Locally Symmetric Spaces

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The Minakshisundaram-Pleijel coefficients for the vector-valued heat kernel on compact locally symmetric spaces of negative curvature

Transactions of the American Mathematical Society ◽

10.1090/s0002-9947-1980-0570777-5 ◽

1980 ◽

Vol 260 (1) ◽

pp. 1-1

Author(s):

Roberto J. Miatello

Keyword(s):

Heat Kernel ◽

Negative Curvature ◽

Symmetric Spaces ◽

Locally Symmetric Spaces ◽

Vector Valued

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An example of an amenable action from geometry

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004016 ◽

1987 ◽

Vol 7 (2) ◽

pp. 289-293 ◽

Cited By ~ 8

Author(s):

R. J. Spatzier

Keyword(s):

Compact Manifold ◽

Negative Curvature ◽

Universal Cover ◽

Constant Negative Curvature ◽

Type Iii ◽

Natural Measure ◽

Amenable Action ◽

Measure Class

AbstractLet M be a compact manifold of not necessarily constant negative curvature. We observe that π1(M) acts amenably on the sphere at infinity of the universal cover of M with respect to a natural measure class. We also note that this action is of type III1.

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