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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board Member: Bannai, Eiichi / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Pasini, Antonio / Ratcliffe, John G. / Scharlau, Rudolf / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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IMPACT FACTOR 2016: 0.552

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1615-7168
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Volume 15, Issue 4 (Oct 2015)

Issues

The Log-Convex Density Conjecture and vertical surface area in warped products

Sean Howe
Published Online: 2015-10-06 | DOI: https://doi.org/10.1515/advgeom-2015-0026

Abstract

We examine the vertical component of surface area in the warped product of a Euclidean interval and a fiber manifold with product density.We determine general conditions under which vertical fibers minimize vertical surface area among regions bounding the same volume and use these results to conclude that in many such spaces vertical fibers are isoperimetric. Our main hypothesis is that the surface area of a fiber be a convex function of the volume it bounds. We apply our results in the specific case of ℝn − {0} realized as the warped product (0, ∞) ×r Sn−1, providing many new examples of densities where spheres about the origin are isoperimetric, including simple densities with finite volume, simple densities that at the origin are neither log-convex nor smooth, and non-simple densities. We also generalize the results of Kolesnikov and Zhdanov on large balls in Rn with increasing strictly log-convex simple density. We situate our work in relation to the Log-Convex Density Conjecture of Rosales et al. and the recent work by Morgan, Ritoré, and others on formulating a generalized log-convex density/stable spheres conjecture.

Keywords: Manifolds with density; Log-Convex Density Conjecture; isoperimetric; warped product; product density; stable spheres

About the article

Received: 2013-11-14

Revised: 2014-01-07

Published Online: 2015-10-06

Published in Print: 2015-10-01


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2015-0026.

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© 2015 by Walter de Gruyter Berlin/Boston. Copyright Clearance Center

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Jiri Dadok and Peter Sternberg
The Journal of Geometric Analysis, 2017
[2]
A. Alvino, F. Brock, F. Chiacchio, A. Mercaldo, and M.R. Posteraro
Journal of Mathematical Analysis and Applications, 2017, Volume 451, Number 1, Page 280

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