(1.1) ut = ∆F (u) on a complete Riemannian manifold (M,g) of dimension n ≥ 1 with Ric(M) ≥ −k for some k ≥ 0. Here F ∈ C2(0,∞), F ′ > 0, and ∆ is the Laplace-Beltrami operator of the metric g. There is a lot of literature on this kind of topics. For example, related problems such as Porous Media Equations have been considered by D.G. Aronson [1], G. Auchmuty and D. Bao [2], M.A. Herrero and M. ...

We present a method for numerical computation of period integrals rigid Calabi-Yau threefold using Picard-Fuchs operator one-parameter smoothing. Our gives possibility computing the lattice double octic without any explicit knowledge its geometric properties, employing only simple facts from theory Fuchsian equations and computations in MAPLE with library differential equations. As surprising c...

We describe some recent ‘oscillation’ estimates in geometric PDE, where estimates are produced using the maximum principle applied to functions depending on several points. Applications include sharp short-time regularity results, sharp long-time behaviour which related closely to optimal estimates on eigenvalues, and elegant proofs of several key results on geometric evolution equations. 1. A ...

Let x : M → Sn+1(1) be an n-dimensional compact hypersurface with constant scalar curvature n(n − 1)r, r ≥ 1, in a unit sphere Sn+1(1), n ≥ 5, and let Js be the Jacobi operator of M . In 2004, L. J. Aĺıas, A. Brasil and L. A. M. Sousa studied the first eigenvalue of Js of the hypersurface with constant scalar curvature n(n− 1) in Sn+1(1), n ≥ 3. In 2008, Q.-M. Cheng studied the first eigenvalue...

The naive approach is to parametrize these deformations as the zero-set of a “mean curvature operator”, then study them using the implicit function theorem. However, this entails a good understanding of the Jacobi operator of the initial submanifold Σ, which in general is not possible. The work of Oh and, more recently, of McLean (cfr. [Oh], [ML]) shows that, in the “right” geometric context, t...

The goal of this paper is to make the vertex operator algebra approach to mirror symmetry accessible to algebraic geometers. Compared to betterknown approaches using moduli spaces of stable maps and special Lagrangian fibrations, this approach follows more closely the original line of thinking that lead to the discovery of mirror symmetry by physicists. The ultimate goal of the vertex algebra a...

In my previous paper [2] with I. M. Singer, B. Wong and Stephen Yau, I gave a lower estimate of the gap of the first 2 eigenvalues of the Schrödinger operator in case the potential is convex. In this note we note that the estimate can be improved if we assume the potential is strongly convex. In particular if the Hessian of the potential is bounded from below by a positive constant, the gap has...

Our purpose in this paper is to study of the eigenvalues {?i(?)}i Dirichlet problem(??)s1u=?((??)s2u+?u)in?,u=0inRN??, where 02s1 and (??)s fractional Laplacian operator defined principle value sense. We first show existence a sequence eigenvalues, which approaches infinity. Secondly we provide Berezin–Li–Yau type lower bound for sum above problem. Furthermore, using self-contained ...

For certain subgroups of $M_{24}$, we give vertex operator algebraic module constructions whose associated trace functions are meromorphic Jacobi forms. These forms canonically to the mock modular Mathieu moonshine. The construction is related Conway moonshine and employs a technique introduced by Anagiannis--Cheng--Harrison. With this able concrete realizations cuspidal Hecke eigenforms weight...