Abstract. This paper studies, using the Bochner technique, a sharp lower bound of the first eigenvalue of a subelliptic Laplace operator on a strongly pseudoconvex CR manifold in terms of its pseudo-Hermitian geometry. For dimensions greater than or equal to 7, the lower bound under a condition on the Ricci curvature and the torsion was obtained by Greenleaf. We give a proof for all dimensions greater than or equal to 5. For dimension 3, the sharp lower bound is proved under a condition which also involves a distinguished covariant derivative of the torsion.
In the paper, we provide an explicit formula for computing the Webster pseudo Ricci curvature, we also apply this formula to obtain a theorem on characterizing balls by using area and pseudo scalar curvature.
Let f : (C n+1 , 0) → (C, 0) be a holomorphic germ defining an isolated hypersurface singularity V at the origin. Let µ and ν and p g be the Milnor number, multiplicity and geometric genus of (V, 0), respectively. We conjecture that µ ≥ (ν − 1) n+1 and the equality holds if and only if f is a semi-homogeneous function. We prove that this inequality holds for n = 1, and also for n = 2 or 3 with additional assumption that f is a quasihomogeneous function. For n = 1, if V has at most two irreducible branches at the origin, or if f is a quasi-homogeneous function, then µ = (ν − 1) 2 if and only if f is a homogeneous polynomial. For n = 2, if f is a quasihomogeneous function, then µ = (ν − 1) 3 iff 6p g = ν(ν − 1)(ν − 2) iff f is a homogeneous polynomial after biholomorphic change of variables. For n = 3, if f is a quasi-homogeneous function, then µ = (ν − 1) 4 iff 24p g = ν(ν − 1)(ν − 2)(ν − 3) iff f is a homogeneous polynomial after biholomorphic change of variables.
This paper gives an explicit formula for calculating the Webster pseudo torsion for a strictly pseudoconvex pseudo-hermitian hypersurface. As applications, we are able to classify some pseudo torsion-free hypersurfaces, which include real ellipsoids.
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