Jaigyoung Choe

The isoperimetric inequality for minimal surfaces in a Riemannian manifold

Abstract It is proved that every minimal surface with one or two boundary components in a simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant K satisfies the sharp isoperimetric inequality 4π A ≦ L2 + K A2. Here equality holds if and only if the minimal surface is a geodesic disk in a surface of constant Gaussian curvature K.

The relative isoperimetric inequality in Cartan-Hadamard 3-manifolds

Abstract We prove that a region D outside a convex set C with smooth boundary in a three-dimensional Cartan-Hadamard manifold M satisfies the relative isoperimetric inequality area , with equality if and only if D is isometric to a Euclidean half ball. We also prove similar sharp inequalities when the sectional curvature of M is bounded above by a negative constant.

Noncommutative Minimal Surfaces

We define noncommutative minimal surfaces in the Weyl algebra, and give a method to construct them by generalizing the well-known Weierstrass representation.

Characterizing the harmonic manifolds by the eigenfunctions of the Laplacian

Abstract The space forms, the complex hyperbolic spaces and the quaternionic hyperbolic spaces are characterized as the harmonic manifolds with specific radial eigenfunctions of the Laplacian.

Higher dimensional Schwarz’s surfaces and Scherk’s surfaces

Higher dimensional generalizations of Schwarz’s P-surface, Schwarz’s D-surface and Scherk’s second surface are constructed as complete embedded periodic minimal hypersurfaces in

On the existence and regularity of fundamental domains with least boundary area

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HIGHER DIMENSIONAL MINIMAL SUBMANIFOLDS GENERALIZING THE CATENOID AND HELICOID

Rn.