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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

Online
ISSN
1435-5345
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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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• Other articles by this author:
Published Online: 2013-11-12 | DOI: https://doi.org/10.1515/crelle-2013-0094

## Abstract

Let $𝔪\in ℕ$, $𝔪\ge 2$, and let ${\left\{{p}_{j}\right\}}_{j=1}^{𝔪}$ be a finite subset of 𝕊2 such that $\stackrel{\to }{0}\in {ℝ}^{3}$ 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 ${\overline{D}}_{1},...,{\overline{D}}_{𝔪}$, such that the Gauss map of S is a homeomorphism onto ${𝕊}^{2}-{\left\{{p}_{j}\right\}}_{j=1}^{𝔪}$ and ${D}_{j}\perp {p}_{j}$, 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.

Revised: 2013-09-09

Published Online: 2013-11-12

Published in Print: 2016-01-01

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,

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