We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian manifold of dimension 2m + 1 ≥ 5. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.

Eigenvalues and capacitors. Let X be a Riemannian manifold and be the Laplace operator on X. It is well-known that if X is compact then the spectrum of is discrete and consists of an increasing sequence {k} k= 1 of the eigenvalues (counted with the multiplicities) where 1= 0 and k as k. Moreover, if n= dim X then Weyls asymptotic formula says that (1.1) k cn

Let M 4n be a complete quaternionic K¨ahler manifold with scalar curvature bounded below by .16n(n + 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1)2. If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume.

In this article, we study local holomorphic isometric embeddings from Bn into BN1 × · · · × BNm with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic map between complex Euclidean spaces by the algebraic extension theorem derived along the lines of Mok, and Mok and Ng. Applying holomorphic continuation and analyzing real analytic subvarieties carefully, we show that each component is either a constant map or a proper holomorphic map between balls. Applying a linearity criterion of Huang, we conclude the total geodesy of non-constant components.

We prove a general maximum principle at infinity for properly immersed minimal surfaces with boundary in R^3. An important corollary of this maximum principle at infinity is the existence of a fixed sized regular neighborhood for any properly embedded minimal surface of bounded curvature.