Complete Minimal Submanifolds Research Articles

An application of Moser iteration to complete minimal submanifolds in a sphere

AbstractWe apply the Moser iteration method to obtain a pointwise bound on the norm of the second fundamental form from a bound on its Ln norm for a complete minimal submanifold in a sphere. As an application we show that a complete minimal submanifold in a sphere with finite total curvature and Ricci curvature bounded away from -∞ must be compact. This complements similar results of Osserman and Oliveira in the case the ambient space is the Euclidean and the hyperbolic space respectively.

On minimal submanifolds in an Euclidean space

AbstractThe concepts “super stable” and “super index” for minimal submanifolds in a Euclidean space are introduced. These concepts coincide with the usual concepts “stable” and “index” when the submanifolds have codimension one. We prove that the only complete super stable minimal submanifolds of finite total scalar curvature and of dimension not less than three in a Euclidean space are affine planes. We also prove that a complete minimal submanifold of dimension larger or equal to three in a Euclidean space with finite super index has finitely many ends. (© 2003 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Transience and capacity of minimal submanifolds

We prove explicit lower bounds for the capacity of annular domains of minimal submanifolds P m in ambient Riemannian spaces N n with sectional curvatures bounded from above. We characterize the situations in which the lower bounds for the capacity are actually attained. Furthermore we apply these bounds to prove that Brownian motion defined on a complete minimal submanifold is transient when the ambient space is a negatively curved Hadamard-Cartan manifold. The proof stems directly from the capacity bounds and also covers the case of minimal submanifolds of dimension m > 2 in Euclidean spaces.

On a minimal Lagrangian submanifold of Cn foliated by spheres.

In this paper, we study a minimal Lagrangian submanifold of C n defined by Harvey and Lawson, which we will call the Lagrangian catenoid because we characterize it locally proving that, besides the Lagrangian subspaces, it is the only minimal Lagrangian submanifold foliated by round (n � 1)-spheres of C n . Also we obtain a global characterization of it among the n-dimensional complete minimal submanifolds of C n with finite total scalar curvature.

Uniqueness of Minimal Submanifolds in Euclidean Space

Let M be a properly immersed n-dimensional complete minimal submanifold in Euclidean space Rn+p of dimension n+p. Let A be the second fundamental form of the immersion, and r the extrinsic distance from the origin. Suppose M has one end and inft supr(x)>t r2(x) |A|2(x) 1.

The second variation of nonorientable minimal submanifolds

Suppose M M is a complete nonorientable minimal submanifold of a Riemannian manifold N N . We derive a second variation formula for the area of M M with respect to certain perturbations, giving a sufficient condition for the instability of M M . Some simple applications are given: we show that the totally geodesic R P 2 \mathbb {R} \mathbb {P}^{2} is the only stable surface in R P 3 \mathbb {R} \mathbb {P}^{3} , and we show the non-existence of stable nonorientable cones in R 4 \mathbb {R}^{4} . We reproduce and marginally extend some known results in the truly non-compact setting.

Minimal submanifolds of hyperbolic spaces via harmonic morphisms

In this paper we give a method for constructing complete minimal submanifolds of the hyperbolic spaces H-m. They are regular fibres of harmonic morphisms from H-m with values in Riemann surfaces.

Nonexistence of some complete stable minimal submanifolds

Nonexistence of some complete stable minimal submanifolds

Pinching Theorems of Simons Type for Complete Minimal Submanifolds in the Sphere

In this paper the author proves some pinching theorems of Simons type for complete minimal submanifolds in the sphere, which generalize the relative results by Simons and Yau.

Pinching theorems of Simons type for complete minimal submanifolds in the sphere

In this paper the author proves some pinching theorems of Simons type for complete minimal submanifolds in the sphere, which generalize the relative results by Simons and Yau.