Frank Morgan

Spaces M with metrics and measures, so-called metric measure spaces or mm spaces (see e.g. Gromov [G]), include very general singular manifolds and weighted graphs. In the smooth case, M is a Riemannian manifold endowed with a smooth positive function or “density” f;  the prescribed measure is just f times the Riemannian volume. In freshman calculus one studies surfaces and solids of revolution via their generating curves and regions in the halfplane {x>0} with density f(x) = 2πx. All quotient manifolds of Riemannian manifolds and homogeneous spaces G/K are Riemannian manifolds with density, and mm spaces were previously called spaces of homogeneous type (see [CW, pp. 587, 591]). Another example, long important to probabilists, is Euclidean space with Gaussian density.

Since varying the density does not alter the topology of a Riemannian manifold, it provides additional flexibility in applying analysis to topology, as in generalizations of splitting theorems by Lichnerowicz [L, 1970], Morse inequalities of Witten [W, 1982], and the proof of the Poincaré Conjecture by Perelman [P, 2003], who begins the body of his paper with manifolds with density before passing to requisite technical refinements.

There are many generalizations of the canonical Riemannian intrinsic curvature to manifolds with density. Corwin et al. [CHHSX] note three different generalizations of Gauss curvature for surfaces needed for three different purposes: generalizing Gauss-Bonnet and asymptotic formulas for the perimeter and area of geodesic balls. Wylie (see Fuller References) provides more. Bakry and Émery [BE] provide an infinite family of generalizations of Ricci curvature with associated formal dimensions. My favorite, the “infinite-dimensional” or “non-dimensional” version, obtained by subtracting from the Riemannian Ricci curvature matrix the Hessian of the logarithm of the density, appeared first in Lichnerowicz [Li, 1970] and yields an generalization of the Levy-Gromov isoperimetric inequality which is easier to prove than the classical Riemannian version (see [M]). In the category of manifolds with density, the round sphere is replaced as the model space by Gauss space, which has the simplifying feature than isoperimetric hypersurfaces are hyperplanes, as first proved about 1975 by Borell [Bor] and Sudakov and Tsirel’son [ST], with uniqueness by Carlen and Kerce [CK, 2001]. Sturm [St] and Lott and Villani [LV] have studied Ricci curvature on more singular spaces using optimal transport.

The Stam-Federbush-Gross Gaussian log-Sobolev inequality ([S, 1959, Eqn. 2.3], [F, 1969, Eqn. (14)], [G1, 1975], [G2, Sect. 6]) is an essential ingredient in Perelman’s [P] proof of Poincaré. (Perelman [P, Rmk. 3.2] remarks that it also follows from his methods; see also Guth [Gu, Appendix 2].) That it follows from the Gaussian isoperimetric inequality was more or less noted by Ledoux [L1, 1994] and nailed by Beckner [B, 1999], who noticed the easy derivation by plugging f = g² into Bobkov’s [Bob, 1997] analytic version of the Gaussian isoperimetric inequality (see [L2, p. 126] and [R, Thm. 3.11]), as I learned from Emanuel Milman.

REFERENCES (for a fuller list see next post)

[BE] D. Bakry and M. Émery, Diffusions Hypercontractive, Séminaire de Probabilités XIX 1983/4, Lecture Notes Math. 1123, Springer, 1985, 177-206. Generalized Ricci curvatures.

[BGL] Dominque Bakry, Ivan Gentil, and Michel Ledoux, Analysis and Geometry of Markov Diffusion operators, Grundlehren der Mathematischen Wissenschaften 348, Springer, Cham, 2014.

[B] William Beckner, Geometric asymptotics and the logarithmic Sobolev inequality, Forum Math. 11 (1999), 105-137.

[Bob]  S. Bobkov, An isoperimetric inequality on the discrete cube and an elementary proof of the isoperimetric inequality in Gauss space, Ann. Prob. 25 (1997), 206-214.

[Bor] Christer Borell, The Brunn-Minkowski inequality in Gauss Space, Invent. Math. 30 (1975) 207-216.

[CK] E. A. Carlen and C. Kerce, On the cases of equality in Bobkov’s inequality and Gaussian rearrangement, Calc. Var. 13 (2001), 1-18.

[CW] R. R. Coifman and G. Weiss, Extensions of Hardy Spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.

[CHHSX] Ivan Corwin, Neil Hoffman, Stephanie Hurder, Vojislav Sesum, Ya Xu, Differential geometry of manifolds with density, Rose-Hulman Und. Math. J. 7 (1) (2006). http://www.rose-hulman.edu/mathjournal/v7n1.php

[F] Paul Federbush, A partially alternate derivation of a result of Nelson, J. Math. Phys. 10 (1969), 50-52.

[G] M. Gromov, Metric structures for Riemannian and non-Riemannian spaces, Reprint of the 2001 English edition, Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2007. Based on the 1981 French original; With appendices by M. Katz, P. Pansu and S. Semmes; Translated from the French by Sean Michael Bates. Also: Isoperimetry of waists and concentration of maps, Geom. Func. Anal. 13 (2003), 178-215.

[G1]  Leonard Gross, Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), no. 4, 1061-1083.

[G2]   Leonard Gross, Logarithmic Sobolev inequalities and contractivity properties of semigroups. Dirichlet forms (Varenna, 1992), 54-88, Lecture Notes in Math. 1563, Springer, Berlin, 1993.

[Gu]  Larry Guth, Metaphors in systolic geometry, ArXiv.org (2010).

[L1]  M. Ledoux, A simple analytic proof of an inequality by P. Buser, Proc. Amer. Math. Soc. 121 (1994), 951-959.

[L2]  M. Ledoux, The geometry of Markov diffusion generators, Ann. Fac. Sci. Toulouse Math. 9 (2000), 305-366.

[Li] André Lichnerowicz, Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650-A653.

[LV] J. Lott and C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Ann. Math. 169 (2009), 903–991; arXiv.org (2006).

[M1] Frank Morgan, Geometric Measure Theory: a Beginner’s Guide. Academic Press, fourth edition, 2009.

[M2] Frank Morgan, Manifolds with Density, Notices Amer. Math. Soc. 52 (2005), 853-858.

[N] E. Nelson, The free Markoff field, J. Funct. Anal. 12 (1973), 211-227.

[P] Grisha Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv.org (2003).

[R] Antonio Ros, The isoperimetric problem, David Hoffman, editor, Global Theory of Minimal Surfaces (Proc. Clay Math. Inst. 2001 Summer School, MSRI), Amer. Math. Soc., 2005, 175-209.

[S] A. J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon, Indo. Control 2 (1959), 101-112, Eqn. 2.3.

[St] Karl-Theodor Sturm. On the geometry of metric measure spaces, I, II. Acta Math. 196 (2006), 65-131 and 133-177.

[ST] V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures, J. Soviet Math. (1978), 9-18 (earlier in Russian).

[W] Edward Witten. Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661-692.

For a fuller list, see next post.