MANFREDO P. do CARMO. Selected Papers

Transcription

1 MANFREDO P. do CARMO Selected Papers

2 Manfredo do Carmo, 2003 Photograph by Suely Lima

3 MANFREDO P. do CARMO Selected Papers Edited by Keti Tenenblat 123

4 Manfredo P. do Carmo Instituto de Matemática PuraeAplicada(IMPA) Rio de Janeiro Brazil Editor Keti Tenenblat Universidade de Brasília Brasilia Brazil ISBN e-isbn DOI / Springer Heidelberg Dordrecht London New York Library of Congress Control Number: Math. Subj. Class (2010): 53A10, 53A30, 53C20, 53C40, 53C42 c Springer-Verlag Berlin Heidelberg 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper Springer is part of Springer Science+Business Media (

5 Selected Works of Outstanding Brazilian Mathematicians Mathematics in Brazil is a very young endeavor. Regular research activities and graduate studies go back to the 1950s, when the biannual Brazilian Mathematical Colloquium was launched and Brazil joined the International Mathematical Union (IMU). The Instituto de Matemática Pura e Aplicada (IMPA) was founded in Creation of the SBM, Sociedade Brasileira de Matemática, would follow, in Half a century later, Brazil has reached Group IV in the IMU s ranking and is responsible for about 1.8% of the world s mathematical production (according to MathSciNet). The Brazilian mathematical community is growing steadily, and has earned international reputation in several fields of research. While many great challenges remain ahead, one can certainly be proud of what has been achieved in such a short time. This progress was largely due to a number of leading mathematicians who, in addition to their own fundamental contributions to Mathematics, created genuine schools around them that raised entire research fields in Brazil to the best international standards. Through this collection, the SBM and IMPA aim to highlight their scientific trajectories and to pay tribute to their pioneering efforts. Editors Hilário Alencar César Camacho André Nachbin Marcelo Viana v

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7 Preface The mathematical contribution of Manfredo P. do Carmo has had a lasting impact on several topics of Differential Geometry and especially on minimal and constant mean curvature submanifolds. This volume contains a selection of do Carmo s papers in chronological order. It includes papers with important results in the following areas: relations between topology and curvature of Riemannian manifolds, convexity and rigidity of isometric immersions, minimal submanifolds, in particular, minimal surfaces, conformal immersions, hypersurfaces of constant mean curvature and constant r-mean curvatures and Hopf-type theorems. The volume starts with his commentaries on his scientific life and on the mathematical contribution of the selected papers. The complete lists of his publications, up to 2010, and of his 27 D.Sc. students are included at the end of this volume. The list of publications extends to more than 80 research articles, with a total of 31 co-authors, and several books. Those of us who were his students remember his enthusiasm in his lectures on Differential Geometry courses. They were stimulating to students and they originated the first versions of his books. Although, obviously not reproduced in this volume, some of do Carmo s books have become classics. The book Differential Geometry of Curves and Surfaces has become a textbook for students all over the world, with translations into Spanish, German, Chinese and Portuguese, and it has helped train several generations of geometers, since it was first published in His dedication to his students and to the development of Geometry was of fundamental importance for the expansion of research on Differential Geometry in Brazil. An account of the evolution of this expansion can be found in one of his papers in this volume. Manfredo P. do Carmo continues contributing to Geometry, with enthusiasm. I am privileged to have been his first student, having been introduced to Differential Geometry through his lectures. Brasília Keti Tenenblat vii

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9 Acknowledgments The Brazilian Mathematical Society (SBM), the Institute for Pure and Applied Mathematics (IMPA) and Springer-Verlag GmbH, express their gratitude to the following copyright holders for granting permission to reprint these contributions: American Mathematical Society Do Carmo, M. P.; Lawson, B. Spherical images of convex surfaces, Proc. Amer. Math. Soc., 31 (1972), Do Carmo, M. P.; Peng, C. K. Stable minimal surfaces in R 3 are planes, Bull. Amer. Math. Soc., New Series, 1 (1979), Do Carmo, M. P.; Dajczer, M. Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc., 277 (1983), Do Carmo, M. P.; Dajczer, M.; Mercuri, F. Compact conformally flat hypersurfaces, Trans. Amer. Math. Soc., 288 (1985), Alencar, H.; Do Carmo, M. P. Hypersurfaces with constant mean curvatures in spheres, Proc. Amer. Math. Soc., 120 (1994), Do Carmo, M. P.; Zhou, D. Eigenvalues estimates on complete noncompact Riemannian manifolds and applications, Trans. Amer. Math. Soc., 35 (1999), Verlag Walter De Gruyter Alencar, H.; do Carmo, M.; Elbert, M.F. Stability of hypersurfaces with vanishing r-mean curvature in euclidean space, J. Reine Angew. Math., 554 (2003), do Carmo, M.; Fernández, I. A Hopf theorem for open surfaces in product spaces. Forum Mathematicum, v. 21, p , ix

10 Acknowledgments European Mathematical Society Publishing House Do Carmo, M. P.; Gomes J.M.; G. Thorbergsson, G. The influence of the boundary behaviour on hypersurfaces with constant mean curvature in H nc1, Comm. Math. Helv., 61 (1986), Do Carmo, M. P.; Ritore, M.; Ros, A. Compact minimal hypersurfaces of index one in the real projective space, Comment. Math. Helv., 75 (2000), Finnish Academy of Sciences and Letters Do Carmo, M. P. Minimal Surfaces: Stability and Finiteness, ICM Helsinki 1978, International Mathematical Union, (1978), International Press do Carmo, M.; Warner, F. Rigidity and convexity of hypersurfaces in spheres, J. of Diff. Geom. 4 (1970), Alencar, H.; Do Carmo, M. P.; Tribuzy, R. A theorem of Hopf and the Cauchy Riemann inequality. Communications in Analysis and Geometry, 15 (2007), Alencar, H.; Do Carmo, M. P.; Tribuzy, R. A Hopf theorem for ambient spaces of dimensions higher than three, J. Diff. Geom., 84 (2010), The John Hopkins University Press Do Carmo, M. P.; Dajczer, M. Conformal rigidity, Amer. J. of Math., 109 (1987), Barbosa, J.L.; Do Carmo, M. P. On the size of a stable minimal surface in R 3, Amer. J. of Math., 98 (1976), Princeton University and the Institute for Advanced Study Do Carmo, M. P. The cohomology ring of certain Kählerian manifolds, Ann. of Math., 81 (1965), Do Carmo, M. P.; Wallach, N. Minimal immersions of spheres into spheres, with N. Wallach, Ann. of Math., 93 (1971), x

11 Contents A Summary of the Scientific Activities Summary of the Papers in this Volume... 7 The cohomology ring of certain Kählerian manifolds Ann. Math., 81, 1 14 (1965) Isometric immersions with semi-definite second quadratic forms E. Lima, Archiv der Math., 20, (1969) Minimal submanifolds of a sphere with second fundamental forms of constant length S.S. Chern and S. Kobayashi, in Functional Analysis and related fields, Springer-Verlag, 1970, pp Rigidity and convexity of hypersurfaces in spheres F. Warner, J. Diff. Geom. 4, (1970) A brief survey of minimal submanifolds II Differential Geometrie in Grossem. Klingenberg (editor), Bibliographisches Inst. Zürich, 1971, pp Minimal immersions of spheres into spheres N. Wallach, Ann. Math., 93, (1971) Spherical images of convex surfaces B. Lawson, Proc. Amer. Math. Soc., 31, (1972) On the size of a stable minimal surface in R J.L. Barbosa, Amer. J. Math., 98, (1976) On minimal immersions with parallel normal curvature tensor A.G. Colares, Springer Math. Lect. Notes, 597, (1977) xi

12 Contents Minimal Surfaces: Stability and Finiteness ICM Helsinki 1978, International Mathematical Union, 1978, pp A proof of the general isoperimetric inequality for surfaces J.L. Barbosa, Math. Z. 162, (1978) Stable minimal surfaces in R 3 are planes C.K. Peng, Bull. A.M.S., New Series, 1, 1979, pp Stability of minimal surfaces and eigenvalues of the Laplacian J.L. Barbosa, Math. Z. 173, (1980) Stability of minimal submanifolds Global Differential Geometry and Global Analysis, Berlin, Springer Math. Lec. Notes 838, 1981, pp Rotation hypersurfaces in spaces of constant curvature M. Dajczer, Trans. Amer. Math. Soc., 277, (1983) Stability of hypersurfaces with constant mean curvature J.L. Barbosa, Math. Z., 185, (1984) Compact conformally flat hypersurfaces M. Dajczer and F. Mercuri, Trans. Amer. Math. Soc., 288, (1985) The influence of the boundary behaviour on hypersurfaces with constant mean curvature in H nc J.M. Gomes and G. Thorbergsson, Comm. Math. Helv., 61, (1986) Conformal rigidity M. Dajczer, Amer. J. Math., 109, (1987) Stability of hypersurfaces of Riemannian manifolds with constant mean curvature J.L. Barbosa and J. Eschenburg, Math. Z., 197, (1988) Hypersurfaces of constant mean curvature Differential Geometry, Peñiscola, Spain, Springer Math. Lect. Notes, 1410, pp Stable hypersurfaces with constant scalar curvature H. Alencar and A.G. Colares, Mathematische Z., 213, (1993) Hypersurfaces with constant mean curvatures in spheres H. Alencar, Proc. Amer. Math. Soc., 120, (1994) xii

13 Contents Complete hypersurfaces with constant mean curvature and finite total curvature P. Berard and W. Santos, Ann. Glob. Anal. Geom., 16, (1998) Research on Differential Geometry in Brazil Matemática Contemporânea, 17 (1999), 1 28 Eigenvalues estimates on complete noncompact Riemannian manifolds and applications D. Zhou, Trans. Amer. Math. Soc., 35, (1999) Compact minimal hypersurfaces of index one in the real projective space M. Ritore and A. Ros, Comment. Math. Helv., 75, (2000) Ricci curvature and the topology of open manifolds C. Xia, Math. Ann., 316, (2000) Stability of hypersurfaces with vanishing r-mean curvature in euclidean space H. Alencar and M.F. Elbert, J. Reine Angew. Math., 554, (2003) A theorem of H. Hopf and the Cauchy-Riemann inequality H. Alencar and R. Tribuzy, Commun. Anal. Geom., 15, (2007) A Hopf theorem for open surfaces in product spaces I. Fernandez, Forum Math., 21, (2009) A Hopf theorem for ambient spaces of dimensions higher than three H. Alencar and R. Tribuzy, J. Diff. Geom., 84, 1 17 (2010) Complete List of Publications of Manfredo P. do Carmo List of D.Sc. Students of Manfredo P. do Carmo xiii

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