By the integral method, we prove that any space-like entire graphic self-shrinking solution to Lagrangian mean curvature flow in $\mathbb{R}^{2n}_{n}$ with the indefinite metric $\sum_i dx_idy_i$ is flat. This result improves the previous ones in [9] and [1] by removing the additional assumption in their results. In a similar manner, we reprove its Euclidean counterpart which is established in [1].

We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.

Evolving smooth, compact hypersurfaces in Rn+1 with normal speed equal to a positive power k of the mean curvature improves a certain ‘isoperimetric difference’ for k > n.1. As singularities may develop before the volume goes to zero, we develop a weak level-set formulation for such flows and show that the above monotonicity is still valid. This proves the isoperimetric inequality for n 6 7. Extending this to complete, simply connected 3-dimensional manifolds with nonpositive sectional curvature, we give a new proof for the Euclidean isoperimetric inequality on such manifolds.

Necessary and sufficient conditions for the discreteness of the Laplacian on a noncompact Riemannian manifold M are established in terms of the isocapacitary function of M. The relevant capacity takes a different form according to whether M has finite or infinite volume. Conditions involving the more standard isoperimetric function of M can also be derived, but they are only sufficient in general, as we demonstrate by concrete examples.

In this paper, we present two kinds of total Chern forms c(E,G) and c(E,G) as well as a total Segre form c(E,G) of a holomorphic Finsler vector bundle c(E,G) expressed by the Finsler metric c(E,G), which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in <i>Finsler Geometry</i>, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133144] to some extent. As some applications, we show that the signed Segre forms c(E,G) are positive c(E,G)-forms on c(E,G) when c(E,G) is of positive Kobayashi curvature; we prove, under an extra assumption, that a FinslerEinstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikous one [Finsler geometry on complex vector bundles, in <i>A Sampler of RiemannFinsler Geometry</i>, MSRI Publications, Vol. 50 (Cambridge