Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary

Author(s)
Gabriel Katz

ABSTRACT

As has been observed by Morse [1], any generic vector field*v* on a compact smooth manifold *X* with boundary gives rise to a stratification of the boundary by compact submanifolds , where . Our main observation is that this stratification re-flects the stratified convexity/concavity of the boundary with respect to the *v*-flow. We study the behavior of this stratification under deformations of the vector field *v*. We also investigate the restrictions that the existence of a convex/concave traversing *v*-flow imposes on the topology of *X*. Let be the orthogonal projection of on the tangent bundle of . We link the dynamics of theon the boundary with the property of in *X* being convex/concave. This linkage is an instance of more general phenomenon that we call “holography of traversing fields”—a subject of a different paper to follow.

As has been observed by Morse [1], any generic vector field

Cite this paper

Katz, G. (2014) Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary.*Applied Mathematics*, **5**, 2823-2848. doi: 10.4236/am.2014.517270.

Katz, G. (2014) Stratified Convexity & Concavity of Gradient Flows on Manifolds with Boundary.

References

[1] Morse, M. (1929) Singular Points of Vector Fields under General Boundary Conditions. American Journal of Mathematics, 51, 165-178.

http://dx.doi.org/10.2307/2370703

[2] Goresky, M. and MacPherson, R. (1983) Stratified Morse Theory. Proceedings of Symposia in Pure Mathematics, 40, 517-533.

[3] Goresky, M. and MacPherson, R. (1983) Morse Theory for the Intersection Homology Groups, Analyse et Topologie sur les Espaces Singulieres. Astérisque, Société Mathématique de France, 101, 135-192.

[4] Goresky, M. and MacPherson, R. (1989) Stratified Morse Theory. Springer Verlag, New York. Ergebnisse Vol. 14. Also Translated into Russian and Published by MIR Press, Moscow, 1991.

[5] Katz, G. (2009) Convexity of Morse Stratifications and Spines of 3-Manifolds. JP Journal of Geometry and Topology, 9, 1-119.

[6] Eliashberg, Y. (1970) Singularities of Folding Type. Izvestiya Akad. Nauk SSSR (Math Series), 34, 1110-1126.

[7] Eliashberg, Y. (1972) Surgery of Singularities of Smooth Mappings. Izvestiya Akad. Nauk SSSR (Math Series), 36, 1321-1347.

[8] Katz, G. (2014) Traversally Generic and Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary. arXiv:1407.1345v1 [math.GT]

[9] Golubitsky, M. and Guillemin, V. (1973) Stable Mappings and Their Singularities. Graduate Texts in Mathematics 14. Springer-Verlag, Berlin.

http://dx.doi.org/10.1007/978-1-4615-7904-5

[10] Katz, G. (2009) The Burnside Ring-Valued Morse Formula for Vector Fields on Manifolds with Boundary. Journal of Topology and Analysis, 1, 13-27.

http://dx.doi.org/10.1142/S1793525309000059

[11] tom Dieck, T. (1975) The Burnside Ring of a Compact Lie Group. I. Mathematische Annalen, 215, 235-250.

[12] Gottlieb, D.H. (1996) All the Way with Gauss-Bonnet and the Sociology of Mathematics. American Mathematical Monthly, 103, 457-469.

http://dx.doi.org/10.2307/2974712

[13] Whitney, H. (1937) On Regular Closed Curves in the Plane. Compositio Mathematica, 4, 276-284.

[14] Guth, L. (2009) Minimal Number of Self-Intersections of the Boundary of an Immersed Surface in the Plane. arXiv: 0903.3112v1 [math.DG]

[15] Milnor, J.H. and Stasheff, J.D. (1974) Characteristic Classes. Annals of Mathematics Studies 76. Princeton University Press, Princeton.

[16] Pontriagin, L.S. (1955) Smooth Manifolds and Their Applications in Homotopy Theory. Trudy Matematicheskogo Instituta imeni VA Steklova, 45.

[17] Kosinski, A. (1992) Differential Manifolds. Academic Press, Boston.

[18] Thom, R. (1957-1958) La classification des immersions. Séminaire Bourbaki, 157.

[19] Calabi, E. (1969) Characterization of Harmonic 1-Forms. In: Spencer, D.C. and Iyanaga, S., Eds., Global Analysis: Papers in Honor of K. Kodaira, 101-117.

[20] Farber, M., Katz, G. and Levine, J. (1998) Morse Theory of Harmonic Forms. Topology, 37, 469-483.

http://dx.doi.org/10.1016/S0040-9383(97)82730-9

[21] Perelmann, G. (2002) The Entropy Formula for Ricci Flow and Its Geometric Applications. arXiv: math.DG/0303109

[22] Perelmann, G. (2003) Ricci Flow with Surgery on Three-Manifolds. arXiv: math.DG/0303109

[23] Smale, S. (1962) On the Structure of Manifolds. American Journal of Mathematics, 84, 387-399.

http://dx.doi.org/10.2307/2372978

[24] Benedetti, R. and Petronio, C. (1997) Branched Standard Spines of 3-Manifolds. Lecture Notes in Mathematics 1653, Springer, Berlin.

[25] Matveev, S.M. (1973) Special Spines of Piecewise Linear Manifolds. Matematicheskii Sbornik (N.S.), 92, 282-293.

[1] Morse, M. (1929) Singular Points of Vector Fields under General Boundary Conditions. American Journal of Mathematics, 51, 165-178.

http://dx.doi.org/10.2307/2370703

[2] Goresky, M. and MacPherson, R. (1983) Stratified Morse Theory. Proceedings of Symposia in Pure Mathematics, 40, 517-533.

[3] Goresky, M. and MacPherson, R. (1983) Morse Theory for the Intersection Homology Groups, Analyse et Topologie sur les Espaces Singulieres. Astérisque, Société Mathématique de France, 101, 135-192.

[4] Goresky, M. and MacPherson, R. (1989) Stratified Morse Theory. Springer Verlag, New York. Ergebnisse Vol. 14. Also Translated into Russian and Published by MIR Press, Moscow, 1991.

[5] Katz, G. (2009) Convexity of Morse Stratifications and Spines of 3-Manifolds. JP Journal of Geometry and Topology, 9, 1-119.

[6] Eliashberg, Y. (1970) Singularities of Folding Type. Izvestiya Akad. Nauk SSSR (Math Series), 34, 1110-1126.

[7] Eliashberg, Y. (1972) Surgery of Singularities of Smooth Mappings. Izvestiya Akad. Nauk SSSR (Math Series), 36, 1321-1347.

[8] Katz, G. (2014) Traversally Generic and Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary. arXiv:1407.1345v1 [math.GT]

[9] Golubitsky, M. and Guillemin, V. (1973) Stable Mappings and Their Singularities. Graduate Texts in Mathematics 14. Springer-Verlag, Berlin.

http://dx.doi.org/10.1007/978-1-4615-7904-5

[10] Katz, G. (2009) The Burnside Ring-Valued Morse Formula for Vector Fields on Manifolds with Boundary. Journal of Topology and Analysis, 1, 13-27.

http://dx.doi.org/10.1142/S1793525309000059

[11] tom Dieck, T. (1975) The Burnside Ring of a Compact Lie Group. I. Mathematische Annalen, 215, 235-250.

[12] Gottlieb, D.H. (1996) All the Way with Gauss-Bonnet and the Sociology of Mathematics. American Mathematical Monthly, 103, 457-469.

http://dx.doi.org/10.2307/2974712

[13] Whitney, H. (1937) On Regular Closed Curves in the Plane. Compositio Mathematica, 4, 276-284.

[14] Guth, L. (2009) Minimal Number of Self-Intersections of the Boundary of an Immersed Surface in the Plane. arXiv: 0903.3112v1 [math.DG]

[15] Milnor, J.H. and Stasheff, J.D. (1974) Characteristic Classes. Annals of Mathematics Studies 76. Princeton University Press, Princeton.

[16] Pontriagin, L.S. (1955) Smooth Manifolds and Their Applications in Homotopy Theory. Trudy Matematicheskogo Instituta imeni VA Steklova, 45.

[17] Kosinski, A. (1992) Differential Manifolds. Academic Press, Boston.

[18] Thom, R. (1957-1958) La classification des immersions. Séminaire Bourbaki, 157.

[19] Calabi, E. (1969) Characterization of Harmonic 1-Forms. In: Spencer, D.C. and Iyanaga, S., Eds., Global Analysis: Papers in Honor of K. Kodaira, 101-117.

[20] Farber, M., Katz, G. and Levine, J. (1998) Morse Theory of Harmonic Forms. Topology, 37, 469-483.

http://dx.doi.org/10.1016/S0040-9383(97)82730-9

[21] Perelmann, G. (2002) The Entropy Formula for Ricci Flow and Its Geometric Applications. arXiv: math.DG/0303109

[22] Perelmann, G. (2003) Ricci Flow with Surgery on Three-Manifolds. arXiv: math.DG/0303109

[23] Smale, S. (1962) On the Structure of Manifolds. American Journal of Mathematics, 84, 387-399.

http://dx.doi.org/10.2307/2372978

[24] Benedetti, R. and Petronio, C. (1997) Branched Standard Spines of 3-Manifolds. Lecture Notes in Mathematics 1653, Springer, Berlin.

[25] Matveev, S.M. (1973) Special Spines of Piecewise Linear Manifolds. Matematicheskii Sbornik (N.S.), 92, 282-293.