scholarly journals The Dirichlet problem for the prescribed Ricci curvature equation on cohomogeneity one manifolds

2015 ◽  
Vol 195 (4) ◽  
pp. 1269-1286 ◽  
Author(s):  
Artem Pulemotov
Keyword(s):  
Dirichlet Problem ◽  
Ricci Curvature ◽  
Cohomogeneity One ◽  
Curvature Equation ◽  
Cohomogeneity One Manifolds

Cohomogeneity one manifolds with positive Ricci curvature

2002 ◽  
Vol 149 (3) ◽  
pp. 619-646 ◽  
Author(s):  
Karsten Grove ◽  
Wolfgang Ziller
Keyword(s):  
Ricci Curvature ◽  
Positive Ricci Curvature ◽  
Cohomogeneity One ◽  
Cohomogeneity One Manifolds

scholarly journals SMOOTHNESS CONDITIONS IN COHOMOGENEITY ONE MANIFOLDS

2020 ◽  
Author(s):  
L. VERDIANI ◽  
W. ZILLER
Keyword(s):  
Efficient Method ◽  
Regular Part ◽  
Cohomogeneity One ◽  
Singular Orbit ◽  
Cohomogeneity One Manifolds

Abstract We present an efficient method for determining the conditions that a metric on a cohomogeneity one manifold, defined in terms of functions on the regular part, needs to satisfy in order to extend smoothly to the singular orbit.

scholarly journals The Dirichlet Problem of the Constant Mean Curvature in Equation in Lorentz-Minkowski Space and in Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1211 ◽  
Author(s):  
Rafael López
Keyword(s):  
Dirichlet Problem ◽  
Euclidean Space ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Elliptic Equations ◽  
Constant Mean Curvature ◽  
Euclidean Case ◽  
Curvature Equation ◽  
Mean Curvature Equation ◽  
The Mean

We investigate the differences and similarities of the Dirichlet problem of the mean curvature equation in the Euclidean space and in the Lorentz-Minkowski space. Although the solvability of the Dirichlet problem follows standards techniques of elliptic equations, we focus in showing how the spacelike condition in the Lorentz-Minkowski space allows dropping the hypothesis on the mean convexity, which is required in the Euclidean case.

scholarly journals Complete Ricci Solitons via Estimates on the Soliton Potential

2020 ◽  
Author(s):  
Matthias Wink
Keyword(s):  
Large Class ◽  
Ricci Solitons ◽  
Growth Estimate ◽  
Cohomogeneity One ◽  
Cohomogeneity One Manifolds

Abstract In this paper, a growth estimate on the soliton potential is shown for a large class of cohomogeneity one manifolds. This is used to construct continuous families of complete steady and expanding Ricci solitons in the setups of Lü–Page–Pope [ 24] and Dancer–Wang [ 17]. It also provides a different approach to the two summands system [ 30] that applies to all known geometric examples.

scholarly journals The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

2017 ◽  
Vol 24 (1) ◽  
pp. 113-134 ◽  
Author(s):  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Differential Operators ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Mean Curvature Equations

AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

Cohomogeneity one manifolds and hypersurfaces of the euclidean space

1995 ◽  
Vol 13 (2) ◽  
pp. 169-184 ◽  
Author(s):  
Fabio Podest� ◽  
Andrea Spiro
Keyword(s):  
Euclidean Space ◽  
Cohomogeneity One ◽  
Cohomogeneity One Manifolds
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