We study the class of holomorphic and isometric submersions between finite-type Teichm "uller spaces. We prove that, with potential exceptions coming from low-genus phenomena, any such map is a forgetful map @math obtained by filling in punctures. This generalizes a classical result of Royden and Earle-Kra asserting that biholomorphisms between finite-type Teichm "uller spaces arise from mapping classes. As a key step in the argument, we prove that any @math -linear embedding @math between spaces of integrable quadratic differentials is, up to scale, pull-back by a holomorphic map. We accomplish this step by adapting methods developed by Markovic to study isometries of infinite-type Teichm "uller spaces.
The classification of holomorphic isometric submersions between bounded symmetric domains is an interesting problem. See the paper of Knese @cite_4 for the classification of holomorphic Kobayashi-isometric submersions from the polydisk @math to the disk @math . For Teichm "uller-theoretic applications of this class of functions on the polydisk, see @cite_16 and @cite_5 .
@cite_5 @cite_16 @cite_4
2145814138 2073656615 2041867826 2586093215
We study isometric maps between Teichmuller spaces and bounded symmetric domains in their Kobayashi metric. We prove that every totally geodesic isometry from a disk to Teichmuller space is either holomorphic or anti-holomorphic; in particular, it is a Teichmuller disk. However, we prove that in dimensions two or more there are no holomorphic isometric immersions between Teichmuller spaces and bounded symmetric domains and also prove a similar result for isometric submersions. In the present paper we analyze a class of tensor-structured preconditioners for the multidimensional second-order elliptic operators in ℝ d , d≥2. For equations in a bounded domain, the construction is based on the rank-R tensor-product approximation of the elliptic resolvent ℬ R ≈(ℒ−λ I)−1, where ℒ is the sum of univariate elliptic operators. We prove the explicit estimate on the tensor rank R that ensures the spectral equivalence. For equations in an unbounded domain, one can utilize the tensor-structured approximation of Green’s kernel for the shifted Laplacian in ℝ d , which is well developed in the case of nonoscillatory potentials. For the oscillating kernels e −i κ‖x‖ ‖x‖, x∈ℝ d , κ∈ℝ+, we give constructive proof of the rank-O(κ) separable approximation. This leads to the tensor representation for the discretized 3D Helmholtz kernel on an n×n×n grid that requires only O(κ |log e|2 n) reals for storage. Such representations can be applied to both the 3D volume and boundary calculations with sublinear cost O(n 2), even in the case κ=O(n). We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ with genus γ and k boundary components we obtain the upper bound σ1L(∂Σ)⩽2(γ+k)π. For γ=0 and k=1 this result was obtained by Weinstock in 1954, and is sharp. We attempt to find the best constant in this inequality for annular surfaces (γ=0 and k=2). For rotationally symmetric metrics we show that the best constant is achieved by the induced metric on the portion of the catenoid centered at the origin which meets a sphere orthogonally and hence is a solution of the free boundary problem for the area functional in the ball. For a general class of (not necessarily rotationally symmetric) metrics on the annulus, which we call supercritical, we prove that σ1(Σ)L(∂Σ) is dominated by that of the critical catenoid with equality if and only if the annulus is conformally equivalent to the critical catenoid by a conformal transformation which is an isometry on the boundary. Motivated by the annulus case, we show that a proper submanifold of the ball is immersed by Steklov eigenfunctions if and only if it is a free boundary solution. We then prove general upper bounds for conformal metrics on manifolds of any dimension which can be properly conformally immersed into the unit ball in terms of certain conformal volume quantities. We show that these bounds are only achieved when the manifold is minimally immersed by first Steklov eigenfunctions. We also use these ideas to show that any free boundary solution in two dimensions has area at least π, and we observe that this implies the sharp isoperimetric inequality for free boundary solutions in the two-dimensional case. We study the family of holomorphic maps from the polydisk to the disk which restrict to the identity on the diagonal. In particular, we analyze the asymptotics of the orbit of such a map under the conjugation action of a unipotent subgroup of @math . We discuss an application our results to the study of the Caratheodory metric on Teichmuller space.