no. 7
Mean Curvature in the Light of Scalar Curvature
[Courbure moyenne à la lumière de la courbure scalaire]
Gromov, Misha1, 2
1 Courant Institute of Mathematical Sciences, NYU, New York, (U.S.A)
2 and IHES, Bures-Sur-Yvette (France)
Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3169-3194.

Nous formulons plusieurs conjectures sur les domaines à bords de courbure moyenne positive dans l’espace euclidiens ainsi que dans des espaces plus généraux de courbure scalaire minorée. Nous prouvons quelques théorèmes qui étayent ces conjectures.

We formulate several conjectures on mean convex domains in the Euclidean spaces, as well as in more general spaces with lower bounds on their scalar curvatures, and prove a few theorems motivating these conjectures

Publié le :
DOI : 10.5802/aif.3347
Keywords: mean curvature, scalar curvature
Mot clés : courbure moyenne, courbure scalaire

Gromov, Misha 1, 2

1 Courant Institute of Mathematical Sciences, NYU, New York, (U.S.A)
2 and IHES, Bures-Sur-Yvette (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2019__69_7_3169_0,
     author = {Gromov, Misha},
     title = {Mean {Curvature} in the {Light} of {Scalar} {Curvature}},
     journal = {Annales de l'Institut Fourier},
     pages = {3169--3194},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     doi = {10.5802/aif.3347},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3347/}
}
TY  - JOUR
AU  - Gromov, Misha
TI  - Mean Curvature in the Light of Scalar Curvature
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 3169
EP  - 3194
VL  - 69
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3347/
DO  - 10.5802/aif.3347
LA  - en
ID  - AIF_2019__69_7_3169_0
ER  - 
%0 Journal Article
%A Gromov, Misha
%T Mean Curvature in the Light of Scalar Curvature
%J Annales de l'Institut Fourier
%D 2019
%P 3169-3194
%V 69
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3347/
%R 10.5802/aif.3347
%G en
%F AIF_2019__69_7_3169_0
Gromov, Misha. Mean Curvature in the Light of Scalar Curvature. Annales de l'Institut Fourier, Tome 69 (2019) no. 7, pp. 3169-3194. doi : 10.5802/aif.3347. https://aif.centre-mersenne.org/articles/10.5802/aif.3347/

[1] Basilio, J.; Dodziuk, J.; Sormani, C. Sewing Riemannian manifolds with positive scalar curvature., J. Geom. Anal., Volume 28 (2018) no. 4, pp. 3553-3602 | DOI | MR | Zbl

[2] Goette, S.; Semmelmann, U. Scalar curvature estimates for compact symmetric spaces., Differ. Geom. Appl., Volume 16 (2002) no. 1, pp. 65-78 | DOI | MR | Zbl

[3] Gromov, M. Positive curvature, macroscopic dimension, spectral gaps and higher signatures., Functional analysis on the eve of the 21st century. Volume II. In honor of the eightieth birthday of I. M. Gelfand. Proceedings of a conference, held at Rutgers University, New Brunswick, NJ, USA, October 24-27, 1993, Birkhäuser, 1996, pp. 1-213 | Zbl

[4] Gromov, Mikhael Homotopical effects of dilatation., J. Differ. Geom., Volume 13 (1978), pp. 303-310 | DOI | MR | Zbl

[5] Gromov, Mikhael; Lawson, H. Blaine Jun. The classification of simply connected manifolds of positive scalar curvature., Ann. Math., Volume 111 (1980), pp. 423-434 | DOI | MR | Zbl

[6] Gromov, Mikhael; Lawson, H. Blaine Jun. Spin and scalar curvature in the presence of a fundamental group. I., Ann. Math., Volume 111 (1980), pp. 209-230 | DOI | MR | Zbl

[7] Gromov, Mikhael; Lawson, H. Blaine Jun. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds., Publ. Math., Inst. Hautes Étud. Sci., Volume 58 (1983), pp. 83-196 | DOI | Numdam | Zbl

[8] Gromov, Misha 101 Questions, Problems and Conjectures around Scalar Curvature (https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/positivecurvature.pdf) | Zbl

[9] Gromov, Misha Dirac and Plateau billiards in domains with corners., Cent. Eur. J. Math., Volume 12 (2014) no. 8, pp. 1109-1156 | MR | Zbl

[10] Gromov, Misha Plateau-Stein manifolds., Cent. Eur. J. Math., Volume 12 (2014) no. 7, pp. 923-951 | MR | Zbl

[11] Gromov, Misha Scalar Curvature of Manifolds with Boundaries: Natural Questions and Artificial Constructions (2018) (https://arxiv.org/abs/1811.04311)

[12] Gromov, Misha Four Lectures on Scalar Curvature (2019) (https://arxiv.org/abs/1908.10612v3)

[13] Gromov, Misha Metric Inequalities with Scalar Curvature (2019) (http://www.ihes.fr/~gromov/PDF/Inequalities-July%202017.pdf) | Zbl

[14] Hoffman, D.; Meeks, W. H. III The strong halfspace theorem for minimal surfaces., Invent. Math., Volume 101 (1990) no. 2, pp. 373-377 | DOI | MR | Zbl

[15] Lin, Fang Hua Approximation by smooth embedded hypersurfaces with positive mean curvature, Bull. Aust. Math. Soc., Volume 36 (1987), pp. 197-208 | MR | Zbl

[16] Lohkamp, J. Minimal Smoothings of Area Minimizing Cones (2018) (https://arxiv.org/abs/1810.03157)

[17] López, Francisco J.; Martín, Francisco Complete minimal surfaces in ℝ 3 ., Publ. Mat., Barc., Volume 43 (1999) no. 2, pp. 341-449 | DOI | MR | Zbl

[18] Marques, Fernando C.; Neves, André Rigidity of min-max minimal spheres in three-manifolds., Duke Math. J., Volume 161 (2012) no. 14, pp. 2725-2752 | DOI | MR | Zbl

[19] Schoen, R.; Yau, S. T. Positive Scalar Curvature and Minimal Hypersurface Singularities (2017) (https://arxiv.org/abs/1704.05490)

[20] Sormani, C. Scalar Curvature and Intrinsic Flat Convergence (2016) (https://arxiv.org/abs/1606.08949)

Cité par Sources :