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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2017: 1.49

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Volume 2016, Issue 710


The Minkowski problem, new constant curvature surfaces in ℝ3, and some applications

Antonio Alarcón / Rabah Souam
  • Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, Bâtiment Sophie Germain, Case 7012, 75205 Paris Cedex 13, France
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Published Online: 2013-11-12 | DOI: https://doi.org/10.1515/crelle-2013-0094


Let 𝔪, 𝔪2, and let {pj}j=1𝔪 be a finite subset of 𝕊2 such that 03 lies in its positive convex hull. In this paper we make use of the classical Minkowski problem, to show the complete family of smooth convex bodies 𝒦 in ℝ3 whose boundary surface consists of an open surface S with constant Gauss curvature (respectively, constant mean curvature) and 𝔪 planar compact discs D¯1,...,D¯𝔪, such that the Gauss map of S is a homeomorphism onto 𝕊2-{pj}j=1𝔪 and Djpj, for all j. We derive applications to existence of harmonic diffeomorphisms between domains of 𝕊2, existence of capillary surfaces in ℝ3, and a Hessian equation of Monge–Ampère type.

About the article

Received: 2012-05-24

Revised: 2013-09-09

Published Online: 2013-11-12

Published in Print: 2016-01-01

Funding Source: Vicerrectorado de Política Científica e Investigación de la Universidad de Granada

Funding Source: MCYT-FEDER

Award identifier / Grant number: MTM2007-61775

Funding Source: MCYT-FEDER

Award identifier / Grant number: MTM2011-22547

Funding Source: Junta de Andalucía

Award identifier / Grant number: P09-FQM-5088

Funding Source: MICINN CEI

Award identifier / Grant number: PYR-2012-3 CEI BioTIC GENIL (CEB09-0010)

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2016, Issue 710, Pages 1–19, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2013-0094.

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