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Register a new userThe Global Solutions to Cartans Realization Problem
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We introduce a systematic method to solve a type of Cartans realization problem. Our method builds upon a new theory of Lie algebroids and Lie groupoids with structure group and connection. This approach allows to find local as well as complete solutions, their symmetries, and to determine the moduli spaces of local and complete solutions. We apply our method to the problem of classification of extremal Kahler metrics on surfaces.
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Our main result in this paper is the following: Given $H^m, H^n$ hyperbolic spaces of dimensional $m$ and $n$ corresponding, and given a Holder function $f=(s^1,...,f^{n-1}):partial H^mto partial H^n$ between geometric boundaries of $H^m$ and $H^n$.
Then for each $epsilon >0$ there exists a harmonic map $u:H^mto H^n$ which is continuous up to the boundary (in the sense of Euclidean) and $u|_{partial H^m}=(f^1,...,f^{n-1},epsilon)$.
In this work we consider a question in the calculus of variations motivated by riemannian geometry, the isoperimetric problem. We show that solutions to the isoperimetric problem, close in the flat norm to a smooth submanifold, are themselves smooth and $C^{2,alpha}$-close to the given sub manifold. We show also a version with variable metric on the manifold. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, Allards regularity theorem for minimizing varifolds, the isometric immersion theorem of Nash and a parametric version due to Gromov.
In this paper, we consider the indefinite scalar curvature problem on $R^n$. We propose new conditions on the prescribing scalar curvature function such that the scalar curvature problem on $R^n$ (similarly, on $S^n$) has at least one solution. The key observation in our proof is that we use the bifurcation method to get a large solution and then after establishing the Harnack inequality for solutions near the critical points of the prescribed scalar curvature and taking limit, we find the nontrivial positive solution to the indefinite scalar curvature problem.
The objective of this paper is twofold. First we provide the -- to the best knowledge of the authors -- first result on the behavior of the regular part of the free boundary of the obstacle problem close to singularities. We do this using our second result which is the partial answer to a long standing conjecture and the first partial classification of global solutions of the obstacle problem with unbounded coincidence sets.
We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hu007folder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal paper [Ste89] from 1889 for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present.
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