A Note about Isotopy and Concordance of Positive Scalar Curvature Metrics on Compact Manifolds with Boundary

Alessandro Carlotto ; Chao Li

Symmetry, integrability and geometry: methods and applications, Tome 20 (2024)

Résumé

We study notions of isotopy and concordance for Riemannian metrics on manifolds with boundary and, in particular, we introduce two variants of the concept of minimal concordance, the weaker one naturally arising when considering certain spaces of metrics defined by a suitable spectral “stability” condition. We develop some basic tools and obtain a rather complete picture in the case of surfaces.

Détail

Keywords: positive scalar curvature, isotopy, free boundary minimal surfaces.
Mots-clés : concordance

Bibliographie

[1] Agostiniani V., Mantegazza C., Mazzieri L., Oronzio F., Riemannian Penrose inequality via nonlinear potential theory, arXiv: 2205.11642

[2] Akutagawa K., “An obstruction to the positivity of relative Yamabe invariants”, Math. Z., 243 (2003), 85–98 | DOI | MR | Zbl

[3] Akutagawa K., Botvinnik B., “Manifolds of positive scalar curvature and conformal cobordism theory”, Math. Ann., 324 (2002), 817–840, arXiv: math.DG/0008139 | DOI | MR | Zbl

[4] Akutagawa K., Botvinnik B., “The relative Yamabe invariant”, Comm. Anal. Geom., 10 (2002), 935–969, arXiv: math.DG/0008138 | DOI | MR | Zbl

[5] Bamler R., Kleiner B., Ricci flow and contractibility of spaces of metrics, arXiv: 1909.08710

[6] Bär C., Hanke B., “Boundary conditions for scalar curvature”, Perspectives in Scalar Curvature, v. 2, World Scientific Publishing, Hackensack, NJ, 2023, 325–377 | DOI | MR

[7] Bartnik R., “New definition of quasilocal mass”, Phys. Rev. Lett., 62 (1989), 2346–2348 | DOI | MR

[8] Besse A.L., Manifolds all of whose geodesics are closed, Ergeb. Math. Grenzgeb., 93, Springer, Berlin, 1978 | DOI | MR | Zbl

[9] Besse A.L., Einstein manifolds, Ergeb. Math. Grenzgeb. (3), 10, Springer, Berlin, 1987 | DOI | MR | Zbl

[10] Bourguignon J.-P., “Les variétés de dimension $4$ à signature non nulle dont la courbure est harmonique sont d'Einstein”, Invent. Math., 63 (1981), 263–286 | DOI | MR | Zbl

[11] Bourguignon J.-P., Brezis H., “Remarks on the Euler equation”, J. Funct. Anal., 15 (1974), 341–363 | DOI | MR | Zbl

[12] Bourguignon J.-P., Gauduchon P., “Spineurs, opérateurs de Dirac et variations de métriques”, Comm. Math. Phys., 144 (1992), 581–599 | DOI | MR | Zbl

[13] Bourguignon J.-P., Lawson Jr. H.B., “Stability and isolation phenomena for Yang–Mills fields”, Comm. Math. Phys., 79 (1981), 189–230 | DOI | MR | Zbl

[14] Bray H.L., “Proof of the Riemannian Penrose inequality using the positive mass theorem”, J. Differential Geom., 59 (2001), 177–267 | DOI | MR | Zbl

[15] Cabrera Pacheco A.J., Cederbaum C., “A survey on extensions of Riemannian manifolds and Bartnik mass estimates”, Mexican Mathematicians in the World—Trends and Recent Contributions, Contemp. Math., 775, American Mathematical Society, Providence, RI, 2021, 1–30, arXiv: 1904.05830 | DOI | MR | Zbl

[16] Cabrera Pacheco A.J., Miao P., “Higher dimensional black hole initial data with prescribed boundary metric”, Math. Res. Lett., 25 (2018), 937–956, arXiv: 1505.01800 | DOI | MR | Zbl

[17] Carlotto A., “A survey on positive scalar curvature metrics”, Boll. Unione Mat. Ital., 14 (2021), 17–42 | DOI | MR | Zbl

[18] Carlotto A., Li C., “Constrained deformations of positive scalar curvature metrics”, J. Differential Geom. (to appear) , arXiv: 1903.11772 | MR

[19] Carlotto A., Li C., “Constrained deformations of positive scalar curvature metrics, II”, Comm. Pure Appl. Math., 77 (2024), 795–862, arXiv: 2107.11161 | DOI | MR

[20] Carlotto A., Li C., “Attaching faces of positive scalar curvature manifolds with corners”, Beijing J. Pure Appl. Math. (to appear) , arXiv: 2308.05409

[21] Carlotto A., Wu D., “Contractibility results for certain spaces of Riemannian metrics on the disc”, Math. Res. Lett., 28 (2021), 1033–1045, arXiv: 1908.02475 | DOI | MR | Zbl

[22] Colding T.H., Minicozzi II W.P., A course in minimal surfaces, Grad. Stud. Math., 121, American Mathematical Society, Providence, RI, 2011 | DOI | MR | Zbl

[23] Gromov M., Scalar curvature of manifolds with boundaries: natural questions and artificial constructions, arXiv: 1811.04311

[24] Gromov M., “Mean curvature in the light of scalar curvature”, Ann. Inst. Fourier, 69 (2019), 3169–3194, arXiv: 1812.09731 | DOI | MR | Zbl

[25] Gromov M., Lawson Jr. H.B., “The classification of simply connected manifolds of positive scalar curvature”, Ann. of Math., 111 (1980), 423–434 | DOI | MR | Zbl

[26] Gromov M., Lawson Jr. H.B., “Spin and scalar curvature in the presence of a fundamental group. I”, Ann. of Math., 111 (1980), 209–230 | DOI | MR | Zbl

[27] Huisken G., Ilmanen T., “The inverse mean curvature flow and the Riemannian Penrose inequality”, J. Differential Geom., 59 (2001), 353–437 | DOI | MR

[28] Li C., Mantoulidis C., “Metrics with $\lambda _1(-\Delta + k R) \ge 0$ and flexibility in the Riemannian Penrose inequality”, Comm. Math. Phys., 401 (2023), 1831–1877, arXiv: 2106.15709 | DOI | MR | Zbl

[29] Mantoulidis C., Schoen R., “On the Bartnik mass of apparent horizons”, Classical Quantum Gravity, 32 (2015), 205002, 16 pp., arXiv: 1412.0382 | DOI | MR | Zbl

[30] Miao P., “Positive mass theorem on manifolds admitting corners along a hypersurface”, Adv. Theor. Math. Phys., 6(2002) (2003), 1163–1182, arXiv: math-ph/0212025 | DOI | MR

[31] Rosenberg J., “Manifolds of positive scalar curvature: a progress report”, Surv. Differ. Geom., 11, International Press, Somerville, MA, 2007, 259–294 | DOI | MR | Zbl

[32] Rosenberg J., Stolz S., “Metrics of positive scalar curvature and connections with surgery”, Surveys on Surgery Theory, v. 2, Ann. of Math. Stud., 149, Princeton University Press, Princeton, NJ, 2001, 353–386 | DOI | MR | Zbl

[33] Rosenberg J., Weinberger S., Positive scalar curvature on manifolds with boundary and their doubles, arXiv: 2201.01263

[34] Schoen R., Yau S.T., “On the structure of manifolds with positive scalar curvature”, Manuscripta Math., 28 (1979), 159–183 | DOI | MR | Zbl