We deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid, the elliptic hyperboloid in R implies that for the cone in R. We also prove a new restriction estimate for any surface in R locally isometric to the plane and of finite type.

We prove the existence of a solution to the Monge-Ampère equation detHess(φ) = 1 on a cone over a thrice-punctured two-sphere. The total space of the tangent bundle is thereby a Calabi-Yau manifold with flat special Lagrangian fibers. (Each fiber can be quotiented to three-torus if the the affine monodromy can be shown to lie in SL(3,Z)⋉ R .) Our method is through Baues and Cortés’s result that...

We generalize McShane’s identity for the length series of simple closed geodesics on a cusped hyperbolic surface [17] to hyperbolic cone-surfaces (with all cone angles ≤ π), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus tw...

We study the problem of the existence and nonexistence of positive solutions to a superlinear second–order divergence type elliptic equation with measurable coefficients −∇ · a · ∇u = u (∗), p > 1, in an unbounded cone–like domain G ⊂ R (N ≥ 3). We prove that the critical exponent p∗(a,G) = inf{p > 1 : (∗) has a positive supersolution at infinity in G } for a nontrivial cone– like domain is alw...

We investigate a Gepner-like superstring model described by a combination of multiple minimal models and an N = 2 Liouville theory. This model is thought to be equivalent to the superstring theory on a singular noncompact Calabi-Yau manifold. We construct the modular invariant partition function of this model, and confirm the validity of an appropriate GSO projection. We also calculate the elli...