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Register a new userRigidity theorems by the logarithmic capacity
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In light of the Suita conjecture, we explore various rigidity phenomena concerning the Bergman kernel, logarithmic capacity, Greens function, and Euclidean distance and volume.
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We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp`ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Holder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.
The comparison theory for the Riccati equation satisfied by the shape operator of parallel hypersurfaces is generalized to semi-Riemannian manifolds of arbitrary index, using one-sided bounds on the Riemann tensor which in the Riemannian case correspond to one-sided bounds on the sectional curvatures. Starting from 2-dimensional rigidity results and using an inductive technique, a new class of gap-type rigidity theorems is proved for semi-Riemannian manifolds of arbitrary index, generalizing those first given by Gromov and Greene-Wu. As applications we prove rigidity results for semi-Riemannian manifolds with simply connected ends of constant curvature.
We prove that if two non-trapping obstacles in $mathbb{R}^n$ satisfy some rather weak non-degeneracy conditions and the scattering rays in their exteriors have (almost) the same travelling times or (almost) the same scattering length spectrum, then they coincide.
Let $es$ be the class of analytic and univalent functions in the unit disk $|z|<1$, that have a series of the form $f(z)=z+ sum_{n=2}^{infty}a_nz^n$. Let $F$ be the inverse of the function $fines$ with the series expansion %in a disk of radius at least $1/4$ $F(w)=f^{-1}(w)=w+ sum_{n=2}^{infty}A_nw^n$ for $|w|<1/4$. The logarithmic inverse coefficients $Gamma_n$ of $F$ are defined by the formula $logleft(F(w)/wright),=,2sum_{n=1}^{infty}Gamma_n(F)w^n$. % In this paper, we determine the logarithmic inverse coefficients bound of $F$ for the class In this paper, we first determine the sharp bound for the absolute value of $Gamma_n(F)$ when $f$ belongs to $es$ and for all $n geq 1$. This result motivates us to carry forward similar problems for some of its important geometric subclasses. In some cases, we have managed to solve this question completely but in some other cases it is difficult to handle for $ngeq 4$. For example, in the case of convex functions $f$, we show that the logarithmic inverse coefficients $Gamma_n(F)$ of $F$ satisfy the inequality [ |Gamma_n(F)|,le , frac{1}{2n} mbox{ for } ngeq 1,2,3 ] and the estimates are sharp for the function $l(z)=z/(1-z)$. Although this cannot be true for $nge 10$, it is not clear whether this inequality could still be true for $4leq nleq 9$.
The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.
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