Abstract
In his groundbreaking work from 2002, Perelman introduced two fundamental monotonic quantities: the reduced volume and the entropy.
While the reduced volume was motivated by the Bishop–Gromov volume comparison applied to a suitably constructed 𝑁-space, which becomes Ricci-flat as
References
[1] V. Agostiniani, L. Mazzieri and F. Oronzio, A Green’s function proof of the positive mass theorem, Comm. Math. Phys. 405 (2024), no. 2, Paper No. 54. 10.1007/s00220-024-04941-8Suche in Google Scholar
[2] W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. 10.2307/1970868Suche in Google Scholar
[3] F. J. Almgren, Jr., 𝑄 valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, Bull. Amer. Math. Soc. (N. S.) 8 (1983), no. 2, 327–328. 10.1090/S0273-0979-1983-15106-6Suche in Google Scholar
[4] H.-D. Cao and X.-P. Zhu, Hamilton–Perelman’s proof of the Poincaré conjecture and the geometrization conjecture, preprint (2006), https://arxiv.org/abs/math/0612069. Suche in Google Scholar
[5] T. H. Colding, New monotonicity formulas for Ricci curvature and applications. I, Acta Math. 209 (2012), no. 2, 229–263. 10.1007/s11511-012-0086-2Suche in Google Scholar
[6] T. H. Colding and W. P. Minicozzi, II, Monotonicity and its analytic and geometric implications, Proc. Natl. Acad. Sci. USA 110 (2013), no. 48, 19233–19236. 10.1073/pnas.1203856109Suche in Google Scholar
[7] T. H. Colding and W. P. Minicozzi, II, On uniqueness of tangent cones for Einstein manifolds, Invent. Math. 196 (2014), no. 3, 515–588. 10.1007/s00222-013-0474-zSuche in Google Scholar
[8] T. H. Colding and W. P. Minicozzi, II, Ricci curvature and monotonicity for harmonic functions, Calc. Var. Partial Differential Equations 49 (2014), no. 3–4, 1045–1059. 10.1007/s00526-013-0610-zSuche in Google Scholar
[9] T. H. Colding and W. P. Minicozzi, II, Parabolic frequency on manifolds, Int. Math. Res. Not. IMRN 2022 (2022), no. 15, 11878–11890. 10.1093/imrn/rnab052Suche in Google Scholar
[10] B. Davey, Parabolic theory as a high-dimensional limit of elliptic theory, Arch. Ration. Mech. Anal. 228 (2018), no. 1, 159–196. 10.1007/s00205-017-1187-zSuche in Google Scholar
[11] B. Davey and M. S. V. Garcia, Variable-coefficient parabolic theory as a high-dimensional limit of elliptic theory, Calc. Var. Partial Differential Equations 63 (2024), no. 2, Paper No. 40. 10.1007/s00526-023-02644-xSuche in Google Scholar PubMed PubMed Central
[12] R. S. Hamilton, Monotonicity formulas for parabolic flows on manifolds, Comm. Anal. Geom. 1 (1993), no. 1, 127–137. 10.4310/CAG.1993.v1.n1.a7Suche in Google Scholar
[13] G. Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990), no. 1, 285–299. 10.4310/jdg/1214444099Suche in Google Scholar
[14] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002), https://arxiv.org/abs/math/0211159. Suche in Google Scholar
[15] C.-C. Poon, Unique continuation for parabolic equations, Comm. Partial Differential Equations 21 (1996), no. 3–4, 521–539. 10.1080/03605309608821195Suche in Google Scholar
[16] P. Price, A monotonicity formula for Yang–Mills fields, Manuscripta Math. 43 (1983), no. 2–3, 131–166. 10.1007/BF01165828Suche in Google Scholar
[17] R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Differential Geom. 17 (1982), no. 2, 307–335. 10.4310/jdg/1214436923Suche in Google Scholar
[18] M. Struwe, On the evolution of harmonic maps in higher dimensions, J. Differential Geom. 28 (1988), no. 3, 485–502. 10.4310/jdg/1214442475Suche in Google Scholar
[19] V. Šverák, Theory of pde, Course Notes, https://www-users.cse.umn.edu/~sverak/course-notes.pdf, 2011. Suche in Google Scholar
[20] T. Tao, Poincaré’s legacies, pages from year two of a mathematical blog. Part II, American Mathematical Society, Providence 2009. Suche in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence
Artikel in diesem Heft
- Frontmatter
- No compact split limit Ricci flow of type II from the blow-down
- Deriving Perelman’s entropy from Colding’s monotonic volume
- Gravitational instantons with 𝑆1 symmetry
- Relatively Anosov groups: finiteness, measure of maximal entropy, and reparameterization
- On the Rapoport--Zink space for GU(2, 4) over a ramified prime
- Generalized Jouanolou duality, weakly Gorenstein rings, and applications to blowup algebras
- On polynomial convergence to tangent cones for singular Kähler–Einstein metrics
- Anomaly flow: Shi-type estimates and long-time existence