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The $L_p$-Minkowski problem with super-critical exponents
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abstract
The $L_p$-Minkowski problem deals with the existence of closed convex hypersurfaces in $\mathbb{R}^{n+1}$ with prescribed $p$-area measures. It extends the classical Minkowski problem and embraces several important geometric and physical applications. The Existence of solutions has been obtained in the sub-critical case $p>-n-1$, but the problem remains widely open in the super-critical case $p<-n-1$. In this paper, we introduce new ideas to solve the problem for all the super-critical exponents. A crucial ingredient in our proof is a topological method based on the calculation of the homology of a topological space of ellipsoids.
Forward citations
Cited by 9 Pith papers
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Capillary curvature images
The authors solve the even capillary L_p-Minkowski problem for -n < p < 1, proving existence of smooth even capillary hypersurfaces with prescribed curvature in the half-space.
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Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
For convex bodies in R^3, bounded L_p qth dual curvature with p in [0,1) and q>2+p forces a uniform diameter upper bound and volume lower bound.
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The capillary $L_p$-Minkowski problem
Existence of smooth convex capillary bodies with prescribed capillary L_p-surface area measure is proved for all p>1, with a symmetry condition needed when 1<p<n+1.
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Uniqueness in the near isotropic Lp dual Minkowski problem
For -1<p<1 and q sufficiently close to n, the near-isotropic Lp dual Minkowski problem on the sphere has a unique solution, with a sharp C0 estimate; the even case covers -1<p<q<min{n,n+p}.
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The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space
Smooth even strictly horospherically convex solutions exist for the horospherical p-Christoffel-Minkowski problem and the new p-shifted Weingarten problem in hyperbolic space for p≥−n, under convexity bounds on f.
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Uniqueness of $S_2$-isotropic solutions to the isotropic $L_p$ Minkowski problem
Under a spectral-gap assumption on the Hilbert-Brunn-Minkowski operator, every S2-isotropic solution of the isotropic Lp Minkowski problem in the supercritical range p<-n is the unit ball.
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Capillary $L_p$ Minkowski Flows
Anisotropic capillary Gauss curvature flows converge to smooth solutions of capillary L_p Minkowski problems for even data with p > -n-1 and for non-even data with p > n+1.
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The capillary Orlicz-Minkowski problem
The capillary Orlicz-Minkowski problem is formulated, but the main existence theorem is unsupported because the initial solution of the continuity method is not admissible under the paper's normalization.
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Minkowski Problems for Geometric Measures
A comprehensive survey that organizes the Minkowski problems of convex geometry into a unified framework based on geometric measures as differentials of global geometric invariants.
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