Mean Curvature In Riemannian Manifold Research Articles

Small spheres with prescribed nonconstant mean curvature in Riemannian manifolds

Given a function f on a smooth Riemannian manifold without boundary, we prove that if p∈M is a non-degenerate critical point of f, then a neighborhood of p contains a foliation by spheres with mean curvature proportional to f. This foliation is essentially unique. The nondegeneracy assumption can be substantially relaxed, at the expense of losing the property that the family of spheres with prescribed mean curvature defines a foliation.

Variational aspects of phase transitions with prescribed mean curvature

We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asymptotics that are sharp to order $o(\varepsilon^2)$ and $C^{2,\alpha}$ estimates on multiplicity-one phase transition layers.

Allen-Cahn approximation of mean curvature flow in Riemannian manifolds I, uniform estimates

We are concerned with solutions to the parabolic Allen-Cahn equation in Riemannian manifolds. For a general class of initial conditions we show nonpositivity of the limiting energy discrepancy. This in turn allows us to prove an almost monotonicity formula (a weak counterpart of Huisken’s monotonicity formula) which gives a local uniform control of the energy densities at small scales. These results will be used in [41] to extend previous important results from [31] in Euclidean space, showing convergence of solutions to the parabolic Allen- Cahn equations to Brakke’s motion by mean curvature in Riemannian manifolds.

Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, II: Brakke's flows

We prove convergence of solutions to the parabolic Allen–Cahn equation to Brakke's motion by mean curvature in Riemannian manifolds with Ricci curvature bounded from below. Our results hold for a general class of initial conditions and extend previous results from [T. Ilmanen, Convergence of the Allen–Cahn equation to the Brakke's motion by mean curvature, J. Differential Geom. 31 (1993) 417–461] even in Euclidean space. We show that a sequence of measures, associated to energy density of solutions of the parabolic Allen–Cahn equation, converges in the limit to a family of rectifiable Radon measures, which evolves by mean curvature flow in the sense of Brakke. A key role is played by nonpositivity of the limiting energy discrepancy and a local almost monotonicity formula (a weak counterpart of Huisken's monotonicity formula) proved in [Allen–Cahn approximation of mean curvature flow in Riemannian manifolds, I, uniform estimates, to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci.; arXiv:1308.0569], to get various density bounds for the limiting measures.

Hypersurfaces with constant anisotropic mean curvature in Riemannian manifolds

We formulate a variational notion of anisotropic mean curvature for immersed hypersurfaces of arbitrary Riemannian manifolds. Hypersurfaces with constant anisotropic mean curvature are characterized as critical points of an elliptic parametric functional subject to a volume constraint. We provide examples of such hypersurfaces in the case of rotationally invariant functionals defined in product spaces. These examples include rotationally invariant hypersurfaces and graphs.

CMC hypersurfaces condensing to Geodesic segments and rays in Riemannian manifolds

We construct examples of compact and one-ended constant mean curvature surfaces with large mean curvature in Riemannian manifolds with axial symmetry by gluing together small spheres positioned end-to-end along a geodesic. Such surfaces cannot exist in Euclidean space, but we show that the gradient of the ambient scalar curvature acts as a ‘friction term’ which permits the usual analytic gluing construction to be carried out.

Surfaces of bounded mean curvature in Riemannian manifolds

Consider a sequence of closed, orientable surfaces of fixed genus $g$ in a Riemannian manifold $M$ with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface $F\subset M$, together with bounds on the geometry of $M$, give an upper bound on the diameter of $F$. Our proof is modelled on Gromov’s compactness theorem for $J$-holomorphic curves.

Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds

Existence of hypersurfaces with prescribed mean curvature in Riemannian manifolds

A Geometric Maximum Principle for Surfaces of Prescribed Mean Curvature in Riemannian Manifolds

A Geometric Maximum Principle for Surfaces of Prescribed Mean Curvature in Riemannian Manifolds

Stability of hypersurfaces of constant mean curvature in Riemannian manifolds

Stability of hypersurfaces of constant mean curvature in Riemannian manifolds