{"id":526,"date":"2011-03-23T08:37:43","date_gmt":"2011-03-23T12:37:43","guid":{"rendered":"http:\/\/blogs.williams.edu\/Morgan\/?p=526"},"modified":"2011-10-21T05:13:24","modified_gmt":"2011-10-21T10:13:24","slug":"stable-immersions-roun","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2011\/03\/23\/stable-immersions-roun\/","title":{"rendered":"Stable Immersions Round"},"content":{"rendered":"

Barbosa and do Carmo [BdC] proved\u00a0that a compact, stable, oriented, immersed constant-mean-curvature surface S in R<\/strong><\/em>3<\/sup> is umbilic and hence a round sphere. <\/em>The proof works for hypersurfaces in R<\/strong>n<\/sup> as well. The proof was simplified by Wente [W], generalized to cones by Morgan and Ritor\u00e9 [MR], incorrectly generalized to warped products by Montiel [M], and generalized to smooth elliptic integrands by Palmer [P]. Tashiro [T] generalized the fact that umbilic hypersurfaces are round. Locally constant normal variations show that stable implies connected.<\/p>\n

Here we give a streamlined version of the proof without passing through the Minkowski formulae.<\/p>\n

Since S<\/em> is compact, we may assume that the (constant) inward mean curvature\u00a0H<\/em> is positive.<\/p>\n

For unit scaling, At<\/sub> = t2<\/sup>A, Vt<\/sub> = t3<\/sup>V;\u00a0when t=1, dA\/dt = 2A, dV\/dt = 3V, A\u2032 = dA\/dV = (2\/3)A\/V.<\/p>\n

For a constant unit normal variation,\u00a0A\u2032 = 2H.<\/p>\n

Equilibrium says that initially A\u2032|scaling =\u00a0 A\u2032|normal, so A = 3VH.<\/p>\n

The stability hypothesis is nonnegative second variation, which implies that<\/p>\n

(*)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 A\u2033|scaling + A\u2033|normal \u20132A\u2033|mixed \u2265 0.<\/p>\n

A\u201d|scaling = (2\/3)((2\/3)(A\/V)V\u2013A)\/V2<\/sup> = \u2013(2\/9)A\/V2<\/sup>.<\/p>\n

By the second variation formula (or the fact that dH\/dt = -.5|\u03c3|2<\/sup>),<\/p>\n

A\u2033|normal = \u2013A\u20132<\/sup>\u222b|\u03c3|2<\/sup> \u2264 \u20132A\u20132<\/sup>H2<\/sup>A = \u2013(2\/9)A\/V2<\/sup>,<\/p>\n

with equality if and only if umbilic (hence round sphere).<\/p>\n

As scaling derivative of normal derivative,<\/p>\n

A\u2033|mixed = \u20132H\/3V =\u00a0 \u2013(2\/9)A\/V2<\/sup> .<\/p>\n

Alternatively, as normal derivative of scaling derivative,<\/p>\n

A\u2033|mixed =\u00a0(2\/3)((2\/3)(A\/V)V\u2013A)\/V2<\/sup> = \u2013(2\/9)A\/V2<\/sup>.<\/p>\n

Hence equality holds in (*) and S<\/em> is a round sphere. QED<\/p>\n

Cones<\/strong> [MR]. In cone C<\/em> over M<\/em>, same except dH\/dt = -.5|\u03c3|2<\/sup> \u2013 Ric,<\/p>\n

A\u2033|normal = \u2013A\u20132<\/sup>\u222b|\u03c3|2<\/sup> + Ric(n,n) \u2264 \u20132A\u20132<\/sup>H2<\/sup>A \u2013A\u20131<\/sup>Ric(n,n) = \u2013(2\/9)A\/V2 <\/sup>\u2013A\u20131<\/sup>Ric(n,n)<\/p>\n

Hence e.g. if RicM<\/sub> > n-1, which means Ric is positive except radially 0, stable hypersurface must be completely tangential. Spheres are of course always marginally stable, indeed invariant, under the normal\/scaling variation, but need not always be for all variations.<\/p>\n

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

[BdC] \u00a0 J. Lucas Barbosa and Manfredo do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984) 339-353.<\/p>\n

[M] \u00a0 Sebasti\u00e1n Montiel, Stable constant mean curvature hypersurfaces in some Riemannian manifolds,\u00a0Comment. Math. Helv. 73 (1998), 584-602. The\u00a0proof is flawed by assuming without justification that \u03c6 = f\u2032 is constant (top of p. 596), which holds just for cones.<\/p>\n

[MR] \u00a0 Frank Morgan and Manuel Ritor\u00e9, Isoperimetric regions in cones, Trans. Amer. Math. Soc. 354 (2002), 2327-2339.<\/p>\n

[P] \u00a0 Bennett Palmer, Stability of the Wulff shape, Proc. AMS 126 (1998) 3661-3667. See also his arXiv post<\/a> (2011) on piecewise smooth surfaces.<\/p>\n

[T] \u00a0Y. Tashiro, Complete Riemannian manifolds and some vectorfields, Trans. Amer. Math. Soc., 117<\/strong> (1965), 251-275.<\/p>\n

[W] \u00a0 Henry C. Wente, A note on the stability theorem of J. L. Barbosa and M. do Carmo for closed surfaces of constant mean curvature, Pacific J. Math 147 (1991) 375-379.<\/p>\n

 <\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

Barbosa and do Carmo [BdC] proved\u00a0that a compact, stable, oriented, immersed constant-mean-curvature surface S in R3 is umbilic and hence a round sphere. The proof works for hypersurfaces in Rn as well. The proof was simplified by Wente [W], generalized to cones by Morgan and Ritor\u00e9 [MR], incorrectly generalized to warped products by Montiel [M], […]<\/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-526","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/526","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=526"}],"version-history":[{"count":2,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/526\/revisions"}],"predecessor-version":[{"id":778,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/526\/revisions\/778"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=526"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=526"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=526"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}