{"id":70,"date":"2009-10-15T09:25:19","date_gmt":"2009-10-15T13:25:19","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=70"},"modified":"2017-08-25T04:26:53","modified_gmt":"2017-08-25T09:26:53","slug":"symmetrization","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2009\/10\/15\/symmetrization\/","title":{"rendered":"Symmetrization"},"content":{"rendered":"

Write-up of a departmental faculty seminar, October 2, 2009.<\/p>\n

Solutions to problems in geometry and physics and even in the social sciences tend to be symmetric. As prime example, the solution to the isoperimetric problem, which seeks the least-perimeter way to enclose given volume in\u00a0R<\/strong>3<\/sup>, is a sphere, the most symmetric of all shapes. One way to prove this is to show that anything else improves as you make it more symmetric. For thousands of years, mathematicians have been looking for good ways to make shapes more symmetric and to prove that as they get more symmetric they \u201cget better,\u201d for example, enclose the same volume with less perimeter.<\/p>\n

My favorite references are Burago and Zalgaller [BZ, \u00a79.2] and Ros [R1, \u00a73.2]. This talk is based on [MHH]. Gromov [G, \u00a79.4] provides some sweeping remarks and generalizations, including most of our results.<\/p>\n

1. Steiner symmetrization <\/strong>[St, 1838] replaces every vertical slice of a region in\u00a0R<\/strong>3<\/sup> with a centered interval of the same length, as in Figure 1. By calculus, the volume does not change, but one can show that the perimeter decreases (or remains the same).
\n\"\"\"\"<\/a><\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Figure 1. Steiner symmetrization replaces every vertical slice with a centered interval of the same length. www.math.utah.edu\/~treiberg\/Lect.html<\/p>\n

<\/p>\n

2. Schwarz Symmetrization<\/strong> [Sc, 1884] replaces every horizontal slice of a region in\u00a0R<\/strong>3<\/sup> with a centered round disc of the same area. Again, volume does not change, but the perimeter decreases or remains the same.<\/p>\n

\"\"\u00a0\"\"\"\"<\/a><\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

 <\/p>\n

Figure 2. Schwarz symmetrization replaces horizontal slices with centered round discs.\u00a0www.toy-library.com.au\/quoits.gif, danbrunskill.blogspot.com\/2009\/03\/imhotep.html, Gatlinburg Trail April 2017<\/p>\n

3. Generalized Schwarz Symmetrization<\/strong>.\u00a0 Steiner and Schwarz symmetrization can be generalized to any product BxR<\/strong>n<\/sup> of a Riemannian manifold B with\u00a0R<\/strong>n<\/sup>, replacing every vertical slice in\u00a0R<\/strong>n<\/sup> of a region with a centered round ball of the same volume. Our two original examples wereR<\/strong>2<\/sup>xR<\/strong>1<\/sup> (Steiner symmetrization) and\u00a0R<\/strong>1<\/sup>xR<\/strong>2<\/sup> (Schwarz symmetrization). More generally, one can replace BxR<\/strong>n<\/sup> with BxS<\/strong>n<\/sup> or BxF for any Riemannian manifold F in which balls about some fixed point are perimeter minimizing.<\/p>\n

4. Warped Products.<\/strong> More generally, symmetrization generalizes to a warped product Bxg<\/sub>F, in which each fiber {b}xF is scaled by a factor g(b), so that the metric takes the form<\/p>\n

ds2<\/sup> = db2<\/sup> + g(b)2<\/sup>dt2<\/sup>.<\/p>\n

A prime example is roughly to view\u00a0R<\/strong>3<\/sup> as a warped product\u00a0R<\/strong>+<\/sup>xb<\/sub>S<\/strong>2<\/sup>, in which the fibers are concentric spheres about the origin parameterized by their intersection with the positive x-axis. Each such spherical fiber is scaled up by its distance b from the origin. Symmetrization replaces the intersection with each such spherical fiber by a spherical cap of the same area centered on the x-axis. This particular symmetrization is called\u00a0spherical symmetrization<\/em>. Spherical symmetrization is traditionally viewed as distinct from Schwarz symmetrization, but now we see that it is just generalized Schwarz symmetrization on a particular warped product.<\/p>\n

\"\"\"\"<\/a><\/p>\n

Figure 3. Spherical symmetrization replaces the intersection with a sphere with a spherical cap. Image by Sean Howe.<\/p>\n

5. Fiber Bundles.<\/strong> More generally, symmetrization generalizes to fiber bundles.\u00a0 By definition, a Riemannian fiber bundle M with base B is a Riemannian manifold together with a \u201cprojection\u201d map P onto another Riemannian manifold B. Each inverse image P–<\/sup>1<\/sup>(p) is called a fiber. We will be interested in the case when the projection is a Riemannian submersion, which means that infinitesimally the restriction of P to the orthogonal complement of a fiber is an isometry, and when each fiber is a scaling of a fixed Riemannian manifold F. The simplest and most famous example is the Hopf fibration P:S<\/strong>3<\/sup>->S<\/strong>2<\/sup> with unit circle fibers. Think of\u00a0S<\/strong>3<\/sup> in\u00a0R<\/strong>4<\/sup> =\u00a0C<\/strong>2<\/sup>, which has isometries of the form<\/p>\n

(z1<\/sub>, z2<\/sub>) -> (ei<\/sup>t<\/sup>z1<\/sub>, ei<\/sup>t<\/sup>z2<\/sub>) ,<\/p>\n

simultaneous identical rotation in each complex line. The fibers of\u00a0S<\/strong>3<\/sup> are the orbits of these isometries. Even locally\u00a0S<\/strong>3<\/sup> is not a Riemannian product B2<\/sup>xS<\/strong>1<\/sup>; instead, nearby fibers spiral around a given fiber. And globally\u00a0S<\/strong>3<\/sup> is not even topologically the same as\u00a0S<\/strong>2<\/sup>xS<\/strong>1<\/sup>.<\/p>\n

Unfortunately there\u2019s no way in general to define symmetrization in the fiber bundle, because there\u2019s no B inside the fiber bundle for centering the symmetrization.* Instead, the symmetrization is defined in the associated warped product Bxg<\/sub>F. It works under the additional hypothesis that infinitesimally horizontal translation scales volume.<\/p>\n

Ros [R2] used fiber bundle symmetrization for Hopf fibrations to solve the isoperimetric problem in certain lens spaces.<\/p>\n

6. Densities.<\/strong> More generally, symmetrization generalizes to densities [M, Chap. 18].<\/p>\n

7. Open Question.<\/strong> Unfortunately, symmetrization does not seem to work for example in\u00a0R<\/strong>2<\/sup> with density exp(r4<\/sup>) = exp((x2<\/sup>+y2<\/sup>)2<\/sup>), because the horizontal translation does not preserve or scale volume. It remains an open conjecture that balls about the origin are isoperimetric. The conjecture applies to any log convex radial density on\u00a0R<\/strong>N<\/sup>. See [RCBM, Conj. 3.12].<\/p>\n

Acknowledgements.<\/strong> This write-up of a departmental faculty seminar is based on joint work with Sean Howe and Nate Harman [MHH], which began with the Williams College, National Science Foundation SMALL undergraduate research Geometry Group at the University of Granada, Spain, summer 2009. We would like to thank Vincent Bayle, Antonio Ca\u00f1ete, Alexander D\u00edaz, Rafael L\u00f3pez, Manuel Ritor\u00e9, Antonio Ros, C\u00e9sar Rosales, and David Thompson for help and inspiration. We acknowledge partial support by the National Science Foundation (grants to Morgan and to the SMALL REU), the Spanish Ministerio de Educaci\u00f3n y Ciencia (grant to Ritor\u00e9, Rosales,\u00a0et al<\/em>.), the University of Granada, and Williams College. We would be grateful for comments on our\u00a0preprint [MHH]<\/a>.<\/p>\n

References<\/strong><\/p>\n

[BZ]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Yu. D. Burago and V. A.\u00a0 Zalgaller, Geometric Inequalities, Springer-Verlag, 1980.<\/p>\n

[G]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal. 13 (2003), 178\u2013215.<\/p>\n

[MHH]\u00a0\u00a0\u00a0\u00a0\u00a0 Frank Morgan, Sean Howe, and Nate Harman, Steiner and Schwarz symmetrization in warped products and fiber bundles with density,\u00a0arXiv.org<\/a> (2009).<\/p>\n

[R1]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Antonio Ros, The isoperimetric problem, Global Theory of Minimal Surfaces (Proc. Clay Math. Inst. Summer School, 2001), Amer. Math. Soc., Providence, RI (2005), 175-207.<\/p>\n

[R2]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Antonio Ros, The isoperimetric problem for lens spaces, in preparation.<\/p>\n

[RCBM]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 C\u00e9sar Rosales, Antonio Ca\u00f1ete, Vincent Bayle, and Frank Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. PDE 31 (2008), 27-46.<\/p>\n

[Sc]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 H. A. Schwarz, Beweis des Satzes, dass die Kugel kleinere Oberfl\u00e4che besitzt, als jeder andere K\u00f6rper gleichen Volumens, Nachrichten K\u00f6niglichen Gesellschaft Wissenschaften G\u00f6ttingen (1884), 1\u201313.<\/p>\n

[St]\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 J. Steiner, Einfache Beweise der isoperimetrischen Hauptsatze, Crelle\u2019s J. 18 (1838), 286-287.<\/p>\n

*Note on Section 5 added March, 2011. Daniel John<\/a> did the best you can to define symmetrization inside symmetric spaces of noncompact type.<\/p>\n

Note<\/a>:\u00a0Steiner and Schwarz symmetrization fail for non-product densities f<\/em> on\u00a0R<\/strong>n<\/em><\/sup>.<\/p>\n

Note: For the ultimate analysis of equality, see Cagnetti et al<\/a>., 2013. Earlier results on spherical symmetrization appear in Morgan-Pratelli [MP, 6.2, 6.4] and Chung et al. [C, 3.8, 3.9].<\/p>\n

[C]\u00a0Ping Ngai Chung, Miguel A. Fernandez, Niralee Shah, Luis Sordo Vieira, Elena Wikner, Are circles isoperimetric in the plane with density e^r?\u00a0preprint (2011).<\/p>\n

[MP]\u00a0Aldo Pratelli, Existence of isoperimetric regions in\u00a0R<\/strong>n<\/sup><\/em>\u00a0<\/strong>with density,\u00a0Ann. Global Anal. Geom.<\/a>\u00a0\u00a043 (2013), 331\u2013365;\u00a0arXiv.org<\/a>\u00a0(2011).<\/p>\n","protected":false},"excerpt":{"rendered":"

Write-up of a departmental faculty seminar, October 2, 2009. Solutions to problems in geometry and physics and even in the social sciences tend to be symmetric. As prime example, the solution to the isoperimetric problem, which seeks the least-perimeter way to enclose given volume in\u00a0R3, is a sphere, the most symmetric of all shapes. One […]<\/p>\n","protected":false},"author":269,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[14043],"tags":[],"class_list":["post-70","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/70","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/users\/269"}],"replies":[{"embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/comments?post=70"}],"version-history":[{"count":16,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/70\/revisions"}],"predecessor-version":[{"id":2663,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/70\/revisions\/2663"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=70"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=70"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=70"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}