We introduce a technique based on Gaussian maps to study whether a surface can lie on a threefold as a very ample divisor with given normal bundle. We give applications, among which one to surfaces of general type and another to Enriques surfaces. In particular, we prove the genus bound g  17 for Enriques Fano threefolds. Moreover we find a new Enriques-Fano threefold of genus 9 whose normalization has canonical but not terminal singularities and does not admit Q-smoothings.

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In this paper, we are concerned with upper bounds of eigenvalues of Laplace operator on compact Riemannian manifolds and finite graphs. While on the former the Laplace operator is generated by the Riemannian metric, on the latter it reflects combinatorial structure of a graph. Respectively, eigenvalues have many applications in geometry as well as in combinatorics and in other fields of mathematics. We develop a universal approach to upper bounds on both continuous and discrete structures based upon certain properties of the corresponding heat kernel. This approach is perhaps much more general than its realization here. Basically, we start with the following entries:(1) an underlying space M with a finite measure+;(2) a well-defined Laplace operator 2 on functions on M so that 2 is a self-adjoint operator in L2 (M,+) with a discrete spectrum;(3) if M has a boundary, then the boundary condition should be chosen so that it does not disrupt self-adjointness of 2 and is of dissipative nature;(4) a distance function dist (x, y) on M so that|{dist| 1 for an appropriate notion of gradient.

Large-time behavior of solutions to the inflow problem of full compressible NavierStokes equations is investigated on the half line \mathbf{R}_+=(0,+\infty). The wave structure, which contains four wavesthe transonic (or degenerate) boundary layer solution, the 1-rarefaction wave, the viscous 2-contact wave, and the 3-rarefaction wave to the inflow problemis described, and the asymptotic stability of the superposition of the above four wave patterns to the inflow problem of full compressible NavierStokes equations is proven under some smallness conditions. The proof is given by the elementary energy analysis based on the underlying wave structure. The main points in the proof are the treatments of the degeneracies in the transonic boundary layer solution and the wave interactions in the superposition wave.