{"id":1464,"date":"2013-07-13T17:22:48","date_gmt":"2013-07-13T22:22:48","guid":{"rendered":"http:\/\/sites.williams.edu\/Morgan\/?p=1464"},"modified":"2013-11-02T13:17:26","modified_gmt":"2013-11-02T18:17:26","slug":"distance-to-boundary-of-manifold-with-density","status":"publish","type":"post","link":"https:\/\/sites.williams.edu\/Morgan\/2013\/07\/13\/distance-to-boundary-of-manifold-with-density\/","title":{"rendered":"Distance to Boundary of Manifold with Density"},"content":{"rendered":"

Jian Ge’s recent ArXiv post<\/a> on “Comparison theorems for manifolds with mean convex boundary,” Theorem 0.1, has a generalization to manifolds with density<\/a>,\u00a0here within a factor of 2 of sharp for constant density:<\/p>\n

Proposition<\/strong>. Consider a smooth complete connected n-dimensional Riemannian manifold with nonempty boundary with smooth (positive) density\u00a0e\u03c8<\/sup>. Suppose that the (n-1) Bakry Emery Ricci curvature<\/em><\/p>\n

Ric –\u00a0 Hess <\/em>\u03c8 – (1\/(n-1))d<\/em>\u03c8@d<\/em>\u03c8<\/p>\n

is at least (n-1)\u03b4 and that the generalized (inward) mean curvature (sum of principal curvatures)\u00a0H = H<\/em>0<\/sub> – \u2202\u03c8\/\u2202n of the boundary satisfies<\/em><\/p>\n

H\/(n-1) \u2265 h,<\/em><\/p>\n

with h > \u221a(2|\u03b4|) if \u03b4\u00a0\u2264 0.\u00a0Then the distance to the boundary is at most<\/em><\/p>\n

\u00a02\/h \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 if \u03b4\u00a0= 0,<\/em><\/p>\n

(\u221a(2\/\u03b4))arccot(h\/\u221a2\u03b4) \u00a0 \u00a0 \u00a0 \u00a0 \u00a0if \u03b4\u00a0> 0,<\/em><\/p>\n

\u00a0(\u221a(2\/|\u03b4|))arccoth(h\/\u221a2|\u03b4|) \u00a0 if \u03b4\u00a0< 0.<\/em><\/p>\n

Proof<\/em> (cf.<\/em> Chapter 18 of my Geometric Measure Theory<\/a> book, 4th ed.). Let dA<\/em> = ef<\/sup><\/em>dA<\/em>0<\/sub> be the element of weighted surface area of the boundary as it flows inward. Then f\u2032<\/em>\u00a0= –H<\/em> and<\/p>\n

f\u2032\u2032 = -II2<\/sup> + \u22022<\/sup>f\/\u2202n2<\/sup> – Ric \u2264 –H<\/em>0<\/sub>2<\/sup>\/(n<\/em>-1) + \u22022<\/sup>\u03c8\/\u2202n2<\/sup> – Ric.<\/p>\n

\u00a0Note that H<\/em>2<\/sup>\u00a0\u2264 2H<\/em>0<\/sub>2<\/sup> + 2(\u2202\u03c8\/\u2202n)2<\/sup>\u00a0 [with strict inequality unless |H<\/em>0<\/sub>| = |\u2202\u03c8\/\u2202n|], so that<\/p>\n

f\u2032\u2032\u00a0\u00a0=\u00a0–H<\/em>2<\/sup>\/2(n<\/em>-1) + (\u2202\u03c8\/\u2202n)2<\/sup>\/(n-1) +\u00a0\u22022<\/sup>\u03c8\/\u2202n2\u00a0<\/sup>– Ric \u2264\u00a0-f\u2032<\/em>2<\/sup>\/2(n<\/em>-1) – (n<\/em>-1)\u03b4<\/em>.<\/p>\n

Let a<\/em> = \u221a(1\/2(n<\/em>-1)), b<\/em> = \u221a((n<\/em>-1)|\u03b4<\/em>|). Since by hypothesis initially a<\/em>f\u2032<\/em><\/em>\u00a0< –b<\/em> when \u03b4<\/em>\u00a0\u2264 0, by the lemma below\u00a0f\u2032<\/em>\u00a0 = –H<\/em> goes to -\u221e and ef<\/sup><\/em>\u00a0goes to 0 as asserted.<\/p>\n

Example<\/em>. For small \u03b5\u00a0> 0, consider a disk of radius 2 about a point O, with Gaussian curvature 1 for 0 \u2264 r<\/em> < \u03b5, Gaussian curvature -1\/2 for \u03b5\u00a0< r<\/em> < 2\u03b5, and flat for 2\u03b5 < r<\/em> \u2264 2. Smooth or relax the smoothness hypotheses slightly. Note that Ge’s Theorem 0.1 does not apply, because the curvature of the boundary, 1\/2, is not greater than 1. Consider a function \u03c8\u00a0with \u03c8(O) = 0 and<\/p>\n

\u00a0\u03c8\u2032(r) = .9r \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 for 0 < r<\/em> <\u00a0\u03b5 ,<\/p>\n

\u03c8\u2032(r) \u00a0= -.9r \u00a0 \u00a0 \u00a0 \u00a0 for \u03b5\u00a0\u00a0< r<\/em> < 2\u03b5 ,<\/p>\n

\u03c8\u2032(r) = 0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0\u00a0for 2\u03b5 < r<\/em> < 2.<\/p>\n

\u00a0For 0 <\u00a0r<\/em>\u00a0<\u00a0\u03b5\u00a0, Ric – Hess \u03c8\u00a0– d\u03c8@d\u03c8 ~ 1 – .9 > 0. For \u03b5\u00a0\u00a0<\u00a0r<\/em>\u00a0< 2\u03b5\u00a0, Ric – Hess\u00a0\u03c8\u00a0– d\u03c8@d\u03c8\u00a0~ -1\/2 + .9 > 0. For 2\u03b5\u00a0<\u00a0r<\/em>\u00a0< 2, Ric – Hess\u00a0\u03c8\u00a0– d\u03c8@d\u03c8\u00a0= 0. Therefore the Proposition applies and implies that the distance to the boundary is always less than 2\/(1\/2) = 4, twice as large as the sharp value of 2.<\/p>\n

Remark.<\/em> There is no such proposition for density alone. The hypotheses<\/p>\n

\u00a0Hess \u03c8\u00a0+ (1\/N<\/em>))d\u03c8@d\u03c8 \u2264 –N<\/em>\u03b4<\/p>\n

\u00a0and on the boundary<\/p>\n

\u00a0\u2202\u03b4<\/em>\/\u2202n < –Nh<\/em><\/p>\n

\u00a0with h<\/em> > \u221a(|\u03b4|)) if \u03b4\u00a0\u2264 0 are incompatible. Given any point, consider the shortest geodesic to the boundary. Along that geodesic, starting from the boundary,<\/p>\n

\u03c8” \u2264 -(1\/N<\/em>)\u03c8’2\u00a0<\/sup> – N<\/em>\u03b4.<\/p>\n

Since by hypothesis initially \u03c8’ < –Nh<\/em>, by the lemma below \u03c8’ would go to -\u221e and the density would go to 0 unless it hits the boundary first, where \u2202\u03c8\/\u2202n = -\u03c8’ would now be greater than Nh<\/em>.<\/p>\n

Here’s a more trivial version for density alone, unrelated to generalized Ricci curvature.<\/p>\n

Proposition<\/strong> (trivial density case). Consider a smooth complete connected n-dimensional Riemannian manifold with nonempty boundary with smooth (positive) density f. Suppose that<\/em><\/p>\n

\u00a0Hess f\u00a0 \u2264 -\u03b4 < 0<\/em><\/p>\n

and that on the boundary<\/em><\/p>\n

\u00a0\u2202f\/\u2202n \u2264 h > 0.<\/em><\/p>\n

\u00a0Then the distance to the boundary is at most h\/\u03b4.<\/em><\/p>\n

Proof<\/em>. Suppose there is a point at distance greater h\/\u03b4<\/em>\u00a0from the boundary. Consider the shortest geodesic to the boundary. By the time the geodesic reaches the point, df<\/em>\/ds<\/em> < 0. If the geodesic never reaches the boundary again, f<\/em> would eventually go negative, a contradiction. By the time the geodesic reaches the boundary again, df\/ds < –h<\/em>, so \u2202f\/\u2202n > h<\/em>, a contradiction of the hypotheses.<\/p>\n

Remark.<\/em> To see that this is sharp, consider density 2-.5x<\/em>2<\/sup> on [-1,1].<\/p>\n

Lemma.<\/strong> Suppose y’\u2264 -a2<\/sup>y2<\/sup> +\/- b2<\/sup> with a > 0, b \u2265 0 and ay(0) < -b except for the – sign case. Then y goes to -\u221e within:<\/em><\/p>\n

-1\/a2<\/sup>y0<\/sub>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0if b = 0,<\/em><\/p>\n

(1\/ab)arccot(-ay0<\/sub>\u00a0\/b) \u00a0 \u00a0 \u00a0 if b > 0,<\/em><\/p>\n

(1\/ab)arccoth(-ay0<\/sub>\/b) \u00a0 \u00a0 if b < 0.<\/em><\/p>\n

\u00a0Proof<\/em>. By comparison with the equality cases y<\/em> = 1\/a2<\/sup><\/em>(x0<\/sub><\/em>+x<\/em>), y<\/em> = (b<\/em>\/a<\/em>)cot(ab<\/em>(x0<\/sub><\/em>+x<\/em>)), y<\/em> = (b<\/em>\/a<\/em>)coth(ab<\/em>(x0<\/sub><\/em>+x<\/em>)). Note that x0<\/sub><\/em>\u00a0is negative; the three functions blow up as x<\/em> approaches 0.<\/p>\n

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

Jian Ge’s recent ArXiv post on “Comparison theorems for manifolds with mean convex boundary,” Theorem 0.1, has a generalization to manifolds with density,\u00a0here within a factor of 2 of sharp for constant density:<\/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-1464","post","type-post","status-publish","format-standard","hentry","category-math"],"acf":[],"_links":{"self":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1464","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=1464"}],"version-history":[{"count":7,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1464\/revisions"}],"predecessor-version":[{"id":1702,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/posts\/1464\/revisions\/1702"}],"wp:attachment":[{"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/media?parent=1464"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/categories?post=1464"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sites.williams.edu\/Morgan\/wp-json\/wp\/v2\/tags?post=1464"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}