Metric Ricci Curvature and Flow for <i>PL</i> Manifolds

Emil Saucan

doi:10.5802/acirm.61

Metric Ricci Curvature and Flow for PL Manifolds

Emil Saucan ¹

¹ Department of Mathematics Technion HAIFA 32000 and Department of Mathematics and Computer Science The Open University of Israel RA’ANANA 43537 ISRAEL

Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 119-129.

Abstract

We summarize here the main ideas and results of our papers [28], [14], as presented at the 2013 CIRM Meeting on Discrete curvature and we augment these by bringing up an application of one of our main results, namely to solving a problem regarding cube complexes.

Published online: 2014-11-12

Zbl: 06938609

DOI: 10.5802/acirm.61

Classification: 51K10, 53C44, 53C21, 65D18, 20F67
Keywords: Wald-Berestovskii curvature, PL manifold, Ricci curvature, surface Ricci flow, Bonnet-Myers Theorem

Author's affiliations:

Emil Saucan ¹

¹ Department of Mathematics Technion HAIFA 32000 and Department of Mathematics and Computer Science The Open University of Israel RA’ANANA 43537 ISRAEL

@article{ACIRM_2013__3_1_119_0,
     author = {Emil Saucan},
     title = {Metric {Ricci} {Curvature} and {Flow} for {\protect\emph{PL}} {Manifolds}},
     journal = {Actes des rencontres du CIRM},
     pages = {119--129},
     publisher = {CIRM},
     volume = {3},
     number = {1},
     year = {2013},
     doi = {10.5802/acirm.61},
     zbl = {06938609},
     language = {en},
     url = {https://acirm.centre-mersenne.org/articles/10.5802/acirm.61/}
}

TY  - JOUR
AU  - Emil Saucan
TI  - Metric Ricci Curvature and Flow for PL Manifolds
JO  - Actes des rencontres du CIRM
PY  - 2013
SP  - 119
EP  - 129
VL  - 3
IS  - 1
PB  - CIRM
UR  - https://acirm.centre-mersenne.org/articles/10.5802/acirm.61/
DO  - 10.5802/acirm.61
LA  - en
ID  - ACIRM_2013__3_1_119_0
ER  -

%0 Journal Article
%A Emil Saucan
%T Metric Ricci Curvature and Flow for PL Manifolds
%J Actes des rencontres du CIRM
%D 2013
%P 119-129
%V 3
%N 1
%I CIRM
%U https://acirm.centre-mersenne.org/articles/10.5802/acirm.61/
%R 10.5802/acirm.61
%G en
%F ACIRM_2013__3_1_119_0

Emil Saucan. Metric Ricci Curvature and Flow for PL Manifolds. Actes des rencontres du CIRM, Volume 3 (2013) no. 1, pp. 119-129. doi : 10.5802/acirm.61. https://acirm.centre-mersenne.org/articles/10.5802/acirm.61/

References
Cited by

[1] S. B. Alexander; R. L. Bishop Comparison theorems for curves of bounded geodesic curvature in metric spaces of curvature bounded above, Differential Geometry and its Applications, Volume 6 (1996) no. 10, pp. 67-86 | DOI | MR | Zbl

[2] E. Appleboim; Y. Hyams; S. Krakovski; C. Sageev; E. Saucan The Scale-Curvature Connection and its Application to Texture Segmentation, Theory and Applications of Mathematics & Computer Science, Volume 3 (2013) no. 1, pp. 38-54 | Zbl

[3] E. Appleboim; E. Saucan; Y. Y. Zeevi Ricci Curvature and Flow for Image Denoising and Superesolution, Proceedings of EUSIPCO 2012 (2012), pp. 2743-2747 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.1749

[4] V. N. Berestovskii Spaces with bounded curvature and distance geometry, Siberian Math. J., Volume 16 (1986) no. 1, pp. 8-19 | DOI | MR

[5] M. Bestvina Geometric group theory and 3-manifolds hand in hand: the fulfillment of Thurston’s vision, Bull. Amer. Math. Soc., Volume 51 (2014) no. 1, pp. 53-70 | DOI | MR | Zbl

[6] L. M. Blumenthal Distance Geometry – Theory and Applications, Claredon Press, Oxford, 1953 | Zbl

[7] L. M. Blumenthal; K. Menger Studies in Geometry, Freeman and Co., San Francisco, CA, 1970 | Zbl

[8] U. Brehm; W. Kühnel Smooth approximation of polyhedral surfaces regarding curvatures, Geometriae Dedicata, Volume 12 (1982) no. 1, pp. 435-461 | MR

[9] Yu. D. Burago; V. A. Zalgaller Isometric piecewise linear immersions of two-dimensional manifolds with polyhedral metrics into $ℝ^{3}$ , St. Petersburg Math. J., Volume 7 (1996) no. 3, pp. 369-385 | Zbl

[10] B. Chow The Ricci flow on the 2-sphere, J. Differential Geom., Volume 33 (1991) no. 2, pp. 325-334 | DOI | MR | Zbl

[11] B. Chow; F. Luo Combinatorial Ricci Flows on Surfaces, J. Differential Geom., Volume 63 (2003) no. 1, pp. 97-129 | DOI | MR | Zbl

[12] J. Giesen Curve Reconstruction, the Traveling Salesman Problem and Menger’s Theorem on Length, Proceedings of the 15th ACM Symposium on Computational Geometry (SoCG) (1999), pp. 207-216 citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.76.1749 | MR | Zbl

[13] M. Gromov; W. Thurston Pinching constants for hyperbolic manifolds, Invent. math., Volume 89 (1987) no. 10, pp. 1-12 | DOI | MR | Zbl

[14] D. X. Gu; E. Saucan Metric Ricci curvature for PL manifolds, Geometry (2013), 12 pages pages | DOI | Zbl

[15] X. D. Gu; S.-T. Yau Computational Conformal Geometry, Advanced Lectures in Mathematics 3, International Press, Somerville, MA, 2008

[16] J. Haantjes Distance geometry. Curvature in abstract metric spaces, Proc. Kon. Ned. Akad. v. Wetenseh., Amsterdam, Volume 50 (1947), pp. 302-314 | MR | Zbl

[17] J. Haantjes A characteristic local property of geodesics in certain spaces, Proc. Kon. Ned. Akad. v. Wetenseh., Amsterdam, Volume 54 (1951), pp. 66-73 | MR | Zbl

[18] Q. Han; J.-X. Hong Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, Mathematical Surveys and Monographs, 130, AMS Providence, RI, 2006

[19] M. Jin; J. Kim; X. D. Gu Discrete Surface Ricci Flow: Theory and Applications, LNCS 4647: Mathematics of Surfaces, Springer-Verlag, Berlin, 2007, pp. 209-232

[20] W. A. Kirk On Curvature of a Metric Space at a Point, Pacific J. Math., Volume 14 (1964), pp. 195-198 | DOI | MR | Zbl

[21] G. Lerman; J. T. Whitehouse On $d$ -dimensional $d$ -semimetrics and simplex-type inequalities for high-dimensional sine functions, Journal of Approximation Theory, Volume 156 (2009) no. 1, pp. 52-81 | DOI | MR | Zbl

[22] K. Menger Zur allgemeinen Kurventheorie, Fundamenta Mathematicae, Volume 10 (1927) no. 1, pp. 96-115 | DOI | Zbl

[23] J. R. Munkres Elementary Differential Topology (rev. ed.), Princeton University Press, Princeton, N.J., 1966 | MR | Zbl

[24] H. Pajot Analytic Capacity, Rectificabilility, Menger Curvature and the Cauchy Integral, LNM 1799, Springer-Verlag, Berlin, 2002, pp. 302-314 | Zbl

[25] C. Plaut Metric Spaces of Curvature $\geq k$ , Handbook of Geometric Topology (Daverman, R. J. and Sher, R. B., editors), 2002, pp. 819-898 | MR

[26] M. Sageev CAT(0) cube complexes and groups, PCMI Lecture Notes (2013), pp. 1-16 (to appear) | Zbl

[27] E. Saucan On a construction of Burago and Zalgaller, The Asian Journal of Mathematics, Volume 16 (2012) no. 4, pp. 587-606 | DOI | MR | Zbl

[28] E. Saucan A Metric Ricci Flow for Surfaces and its Applications, Geometry, Imaging and Computing, Volume 1 (2014) no. 2, pp. 259-301 | DOI | MR | Zbl

[29] E. Saucan; E. Appleboim Curvature Based Clustering for DNA Microarray Data Analysis, LNCS 3523, Springer-Verlag, Berlin, 2005, pp. 405-412

[30] E. Saucan; E. Appleboim Metric Methods in Surface Triangulation, LNCS 5654, Springer-Verlag, Berlin, 2009, pp. 335-355

[31] P. Shvartsman Sobolev $L_{p}^{2}$ -functions on closed subsets of $ℝ^{2}$ , arXiv:1210.0590, Volume 1 (2012) no. 8, to appear pages

[32] D. A. Stone A combinatorial analogue of a theorem of Myers, Illinois J. Math., Volume 20 (1976) no. 1, pp. 12-21 | DOI | MR | Zbl

[33] D. A. Stone Correction to my paper: “A combinatorial analogue of a theorem of Myers”, Illinois J. Math., Volume 20 (1976) no. 1, pp. 551-554 | Zbl

[34] D. A. Stone Geodesics in Piecewise Linear Manifolds, Trans. Amer. Math. Soc., Volume 215 (1976) no. 1, pp. 1-44 | DOI | MR | Zbl

[35] A. Wald Sur la courbure des surfaces, C. R. Acad. Sci. Paris, Volume 201 (1935) no. 8, pp. 918-920 | Zbl

[36] A. Wald Begreudeung einer koordinatenlosen Differentialgeometrie der Flächen, Ergebnisse e. Mathem. Kolloquims, First Series, Volume 7 (1936) no. 8, pp. 24-46 | Zbl

Cited by Sources: