Papers
Topics
Authors
Recent
Search
2000 character limit reached

Some obstructions to positive scalar curvature on a noncompact manifold

Published 20 Feb 2024 in math.DG | (2402.13239v2)

Abstract: We give obstructions for a noncompact manifold to admit a complete Riemannian metric with (nonuniformly) positive scalar curvature. We treat both the finite volume and infinite volume cases.

Authors (1)
Definition Search Book Streamline Icon: https://streamlinehq.com
References (38)
  1. I. Adamson, “ Cohomology theory for non-normal subgroups and non-normal fields”, Proc. Glasgow Math. Assoc. 2, p. 66-76 (1954)
  2. C. Bär and W. Ballmann, “Boundary value problems for elliptic differential operators of first order”, in Surveys in Differential Geometry XVII, International Press, Boston, p. 1-78 (2012)
  3. C. Bär and B. Hanke, “Boundary conditions for scalar curvature”, in Perspectives in scalar curvature, Vol. 2, World Scientific Publishing Co., Hackensack, NJ, p. 325-377 (2023)
  4. S. Braun and R. Sauer, “Volume and macroscopic scalar curvature”, Geom. Funct. Anal. 31, p. 1321-1376 (2021)
  5. M. Bucher, M. Burger, R. Frigerio, A. Iozzi, C. Pagliantini and M. B. Pozzetti, “Isometric embeddings in bounded cohomology”, J. Topol. Anal. 6, p. 1-25 (2014)
  6. S. Cecchini, D. Räde and R. Zeidler. “Nonnegative scalar curvature on manifolds with at least two ends”, J. of Topology ‘6, p. 855-876 (2023)
  7. S. Cecchini and R. Zeidler. “The positive mass theorem and distance estimates in the spin setting”, to appear, Trans. of the AMS
  8. S. Cecchini and R. Zeidler. “Scalar and mean curvature comparison via the Dirac operator”, to appear, Geom. and Top.
  9. S. Chang, S. Weinberger and G. Yu, “Positive scalar curvature and a new index theory for noncompact manifolds”, J. Geom. Phys. 149, 103575, 22 pp. (2020)
  10. J. Chen, P. Liu, Y. Shi and J. Zhu, “Incompressible hypersurface, positive scalar curvature and positive mass theorem”, preprint, https://arxiv.org/abs/2112.14442 (2021)
  11. O. Chodosh and C. Li, “Generalized soap bubbles and the topology of manifolds with positive scalar curvature”, to appear, Annals of Math.
  12. A. Connes, M. Gromov and H. Moscovici, “Conjecture de Novikov et fibrés presque plats”, Comptes Rendus de l’Acad. des Sciences 310, p. 273-277 (1990)
  13. R. Frigerio and C. Pagliantini, “Relative measure homology and continuous bounded cohomology of topological pair”, Pac. J. Math. 257, p. 91-130 (2012)
  14. M. Gromov, “Volume and bounded cohomology”, Publ. Math. de l’IHES 56, p. 5-99(1982)
  15. M. Gromov, “Large Riemannian manifolds”, in Curvature and Topology of Riemannian Manifolds, Lect. Notes in Math. 1201, Springer-Verlag, Berlin, p.108-122 (1986)
  16. M. Gromov, “Plateau-Stein manifolds”, Open Mathematics 12, p. 923-951 (2014)
  17. M. Gromov, “No metrics with positive scalar curvatures on aspherical 5-manifolds”, preprint, https://arxiv.org/abs/2009.05332 (2020)
  18. M. Gromov and B. Lawson, “The classification of simply connected manifolds of positive scalar curvature”, Ann. Math 111, p. 423-434 (1980)
  19. M. Gromov and B. Lawson, “Positive scalar curvature and the Dirac operator on complete Riemannian manifolds”, Publ. Math. IHES 58, p. 83-196 (1983)
  20. B. Hanke and T. Schick, “Enlargeability and index theory”, J. Diff. Geom. 74, p. 293-320 (2006)
  21. B. Hunger “Almost flat bundles and homological invariance of infinite K-area”, New York J. of Math. 25, p. 687-722 (2019)
  22. N. Ivanov, “Leray theorems for l1subscript𝑙1l_{1}italic_l start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norms of infinite chains”, preprint, https://arxiv.org/abs/2012.08690 (2020)
  23. B. Kleiner and J. Lott, “Notes on Perelman’s papers”, Geom. Top. 12, p. 2587-2855 (2008)
  24. Y. Kubota, “Almost flat relative vector bundles and the almost monodromy correspondence”, J. Topol. Anal. 14, p. 353–382 (2022)
  25. C. Löh, “Isomorphisms in l1superscript𝑙1l^{1}italic_l start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT-homology´´, Münster J. of Math. 1, p. 237-266 (2008)
  26. C. Löh and R. Sauer, “Bounded cohomology of amenable covers via classifying spaces”, Enseign. Math. 66, p. 151-172 (2020)
  27. J. Lohkamp. “Scalar curvature and hammocks”, Math. Ann. 313, p. 385-407 (1999)
  28. J. Lott, “Index theory for scalar curvature on manifolds with boundary”, Proc. of the AMS 149, p. 4451-4459 (2021)
  29. A. Mishchenko and N. Teleman, “Almost flat bundles and almost flat structures”, Topol. Methods Nonlinear Anal. 26, p. 75-87 (2005)
  30. J. Rosenberg, “Manifolds of positive scalar curvature: a progress report”, in Surveys in Diff. Geom. 11, Metric and comparison geometry, International Press, Boston, p. 259-294 (2007)
  31. J. Rosenberg and S. Weinberger, “Positive scalar curvature on manifolds with boundary and their doubles”, to appear, Pure and Appl. Math. Quart.
  32. T. Schick and M. Seyedhosseini, “On an index theorem of Chang, Weinberger and Yu”, Münster J. Math. 14, p. 123-154 (2021)
  33. R. Schoen and S.T. Yau, “On the structure of manifolds with positive scalar curvature”, Manuscripta Math. 28, p. 159-183 (1979)
  34. G. Skandalis, “Approche de la conjecture de Novikov par la cohomologie cyclique (d’après A. Connes, M. Gromov et H. Moscovici)”, Séminaire Bourbaki, Astérisque 201-203, p. 299-320 (1991)
  35. A. Song, “A dichotomy for minimal hypersurfaces in manifolds thick at infinity”, to appear, Ann. Sci. Ec. Norm. Sup.
  36. S. Takasu, “Relative homology and relative cohomology theory of groups”, J. Fac. Sci. Univ. Tokyo. 8, p. 75-110 (1959)
  37. X. Wang and W. Zhang, “On the generalized Geroch conjecture for complete spin manifolds”, Chinese Ann. Math. B43, p. 1143-1146 (2022)
  38. S.-T. Yau, “Some function-theoretic properties of complete Riemannian manifold and their applications to geometry”. Indiana Univ. Math. J. 25, p. 659-670 (1976)
Citations (1)
View on Semantic Scholar

Summary

No one has generated a summary of this paper yet.

Sign Up to Summarize

Paper to Video (Beta)

No one has generated a video about this paper yet.

Sign Up to Generate All Videos Subscribe on YouTube

Whiteboard

No one has generated a whiteboard explanation for this paper yet.

Sign Up to Generate

Continue Learning

We haven't generated follow-up questions for this paper yet.

Generate Now

Collections

Sign up for free to add this paper to one or more collections.

Sign Up

Tweets

Sign up for free to view the 3 tweets with 3 likes about this paper.

Sign Up for Free