In this paper, we consider bounded harmonic functions on complete minimal submanifold Mn in RN. When n≥3 and under some conditions, in particulary with ﬁnite total scalar curvature, we obtain that the dimension of the space of bounded harmonic functions is equivalent to the number of ends on Mn.
Department of Mathematics, Zhejiang University
In this paper, we establish the first variational formula and its Euler-Lagrange equation for the total 2p-th mean curvature functional of a n-dimensional submanifold M in a general (n+m)-dimensional Riemannian manifold N. As an example, we prove that closed complex submanifolds in complex projective spaces are critical points of the total 2p-th mean curvature functional, called relatively 2p-minimal submanifolds, for all p. At last, we discuss the relations between relatively 2p-minimal submanifolds and austere submanifolds in real space forms, as well as a special variational problem.
School of Mathematical Sciences,Beijing Normal University,School of Mathematical Sciences,Beijing Normal University