scholarly journals A Lichnerowicz estimate for the first eigenvalue of convex domains in Kähler manifolds

2013 ◽  
Vol 6 (5) ◽  
pp. 1001-1012
Author(s):  
Vincent Guedj ◽  
Boris Kolev ◽  
Nader Yeganefar
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
First Eigenvalue ◽  
Convex Domains ◽  
The First Eigenvalue

Coupled Vortex Equations on Complete Kähler Manifolds

2008 ◽  
Vol 51 (3) ◽  
pp. 467-480
Author(s):  
Yue Wang
Keyword(s):  
Symmetric Spaces ◽  
Existence Results ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Convex Domains ◽  
Hermitian Symmetric Spaces ◽  
Vortex Equations ◽  
Curved Manifolds ◽  
Pseudo Convex ◽  
Simply Connected

AbstractIn this paper, we first investigate the Dirichlet problem for coupled vortex equations. Secondly, we give existence results for solutions of the coupled vortex equations on a class of complete noncompact Kähler manifolds which include simply-connected strictly negative curved manifolds, Hermitian symmetric spaces of noncompact type and strictly pseudo-convex domains equipped with the Bergmann metric.

scholarly journals On relations between principal eigenvalue and torsional rigidity

2020 ◽  
pp. 2050093
Author(s):  
Michiel van den Berg ◽  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
Optimization Problem ◽  
Torsional Rigidity ◽  
Principal Eigenvalue ◽  
Higher Dimensions ◽  
First Eigenvalue ◽  
Thin Domains ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Dimension One

We consider the problem of minimizing or maximizing the quantity [Formula: see text] on the class of open sets of prescribed Lebesgue measure. Here [Formula: see text] is fixed, [Formula: see text] denotes the first eigenvalue of the Dirichlet Laplacian on [Formula: see text], while [Formula: see text] is the torsional rigidity of [Formula: see text]. The optimization problem above is considered in the class of all domains [Formula: see text], in the class of convex domains [Formula: see text], and in the class of thin domains. The full Blaschke–Santaló diagram for [Formula: see text] and [Formula: see text] is obtained in dimension one, while for higher dimensions we provide some bounds.

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