674

Recent Advances in the Geometry of SubmanifoldsDedicated to the Memory of

Franki Dillen (1963–2013)

AMS Special Sessions:Geometry of Submanifolds

October 25–26, 2014: San Francisco State University, CA

Recent Advances in the Geometry of Submanifolds:Dedicated to the Memory of Franki Dillen (1963–2013)

March 14–15, 2015: Michigan State University, East Lansing, MI

Bogdan D. SuceavaAlfonso CarriazoYun Myung Oh

Joeri Van der VekenEditors

American Mathematical Society

Recent Advances in the Geometry of SubmanifoldsDedicated to the Memory of

Franki Dillen (1963–2013)

AMS Special Sessions:Geometry of Submanifolds

October 25–26, 2014: San Francisco State University, CA

Recent Advances in the Geometry of Submanifolds:Dedicated to the Memory of Franki Dillen (1963–2013)

March 14–15, 2015: Michigan State University, East Lansing, MI

Bogdan D. SuceavaAlfonso CarriazoYun Myung Oh

Joeri Van der VekenEditors

Franki Dillen (March 15, 1963 – April 17, 2013)

674

Recent Advances in the Geometry of SubmanifoldsDedicated to the Memory of

Franki Dillen (1963–2013)

AMS Special Sessions:Geometry of Submanifolds

October 25–26, 2014: San Francisco State University, CA

Recent Advances in the Geometry of Submanifolds:Dedicated to the Memory of Franki Dillen (1963–2013)

March 14–15, 2015: Michigan State University, East Lansing, MI

Bogdan D. SuceavaAlfonso CarriazoYun Myung Oh

Joeri Van der VekenEditors

American Mathematical SocietyProvidence, Rhode Island

EDITORIAL COMMITTEE

Dennis DeTurck, Managing Editor

Michael Loss Kailash Misra Catherine Yan

2010 Mathematics Subject Classification. Primary 53A04, 53B25, 53C25, 53C40, 53C42,53C50, 53D12, 53D15, 58C40, 58E35.

Library of Congress Cataloging-in-Publication Data

Names: Dillen, Franki. | Suceava, Bogdan D., 1969– editor.Title: Recent advances in the geometry of submanifolds : dedicated to the memory of Franki

Dillen (1963-2013) : AMS special sessions on geometry of submanifolds, October 25-26, 2014,San Francisco State University, San Francisco, California : recent advances on submanifoldgeometry : dedicated to the memory of Franki Dillen (1963-2013), March 14-15, 2015, Michi-gan State University, East Lansing, Michigan / Bogdan D. Suceava [and three others], editors.

Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Con-temporary mathematics ; volume 674 | Includes bibliographical references.

Identifiers: LCCN 2016003595 | ISBN 9781470422981 (alk. paper)Subjects: LCSH: Submanifolds–Congresses. | Manifolds (Mathematics)–Congresses. | Geom-

etry, Differential–Congresses. | AMS: Differential geometry – Classical differential geometry –Curves in Euclidean space. msc | Differential geometry – Local differential geometry – Localsubmanifolds. msc | Differential geometry – Global differential geometry – Special Riemannianmanifolds (Einstein, Sasakian, etc.). msc | Differential geometry – Global differential geometry– Global submanifolds. msc | Differential geometry – Global differential geometry – Immersions(minimal, prescribed curvature, tight, etc.). msc | Differential geometry – Global differentialgeometry – Lorentz manifolds, manifolds with indefinite metrics. msc | Differential geometry –Symplectic geometry, contact geometry – Lagrangian submanifolds; Maslov index. msc | Differen-tial geometry – Symplectic geometry, contact geometry – Almost contact and almost symplecticmanifolds. msc | Global analysis, analysis on manifolds – Calculus on manifolds; nonlinear op-erators – Spectral theory; eigenvalue problems. msc | Global analysis, analysis on manifolds –Variational problems in infinite-dimensional spaces – Variational inequalities (global problems).msc

Classification: LCC QA649 .R434 2016 | DDC 516.3/62–dc23 LC record available athttp://lccn.loc.gov/2016003595Contemporary Mathematics ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)

DOI: http://dx.doi.org/10.1090/conm/674

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Contents

Preface vii

In memory of Franki Dillen (a biography)Bang-Yen Chen, Joeri Van der Veken, and Luc Vrancken 1

Natural extrinsic geometrical symmetries, an introductionLeopold Verstraelen 5

A survey on semi-Riemannian generalized Sasakian-space-formsAlfonso Carriazo 17

A survey on Ricci solitons on Riemannian submanifoldsBang-Yen Chen 27

The total absolute curvature and the total absolute torsionof open curves in the Euclidean spaces

Kazuyuki Enomoto and Jin-ichi Itoh 41

Vertex-type curves in constant angle surfaces of Hyp2 × R

Brendan Foreman 49

Clelia curves, twisted surfaces and Plucker’s conoid in Euclideanand Minkowski 3-space

Wendy Goemans and Ignace Van de Woestyne 59

Stark hypersurfaces in complex projective spaceThomas A. Ivey 75

Submanifolds related to Gauss map and some differential operatorsYoung Ho Kim 89

The normal Ricci curvature inequalityZhiqin Lu and David Wenzel 99

On the generalized Wintgen inequality for submanifolds in complexand Sasakian space forms

Ion Mihai 111

Some recent progress of biharmonic submanifoldsYe-Lin Ou 127

A nonlinear inequality involving the mean curvature of a spacelike surfacein 3-dimensional GRW spacetimes and Calabi-Bernstein type problems

Alfonso Romero and Rafael M. Rubio 141

v

vi CONTENTS

On Lagrangian submanifolds of the nearly Kaehler 6-sphereRamesh Sharma and Sharief Deshmukh 153

Ideal Lagrangian submanifoldsJoeri Van der Veken 161

Complete Lagrangian ideal δ(2) submanifolds in the complex projective spaceLuc Vrancken 175

Comparison theorems in Riemannian geometry with applicationsShihshu Walter Wei 185

Preface

About a century ago, the geometry of submanifolds gained a lot of momentumthrough the study of the Schlafli’s conjecture, which stated that a real analytic Rie-mannian manifold of dimension n can be locally isometrically embedded into anyreal analytic Riemannian manifold of dimension 1

2n(n+1). M. Janet (1926), E. Car-tan (1927) and C. Burstin (1931) made essential contributions to the understandingof the importance of the immersion problems and to a result that today bears theirnames. A major development for the theory was the much-celebrated EmbeddingTheorem, proved by John Forbes Nash, Jr. (in a series of three papers publishedin 1954, 1956, and 1966). Over the last several decades, many outstanding math-ematicians focused their efforts on the geometry of submanifolds. Notably, FrankiDillen’s work has attracted the attention of and inspired many geometers. Thisis why we thought it appropriate to honor his work in a volume of the AmericanMathematical Society’s Contemporary Mathematics series.

Our aim was to assemble a volume that complements the existing literaturewith new content and new ideas that could serve as inspiration to all mathemati-cians working with concepts related to the geometry of submanifolds. These themesinclude the recent study of submanifolds in Riemannian, semi-Riemannian, Kaehle-rian and contact manifolds. During the last twenty years, the study of new curva-ture invariants (especially the Chen curvature invariants—called by some authorsδ-invariants) inspired techniques that have produced new results. Some of these re-sults have been obtained by using techniques in classical differential geometry, whileothers used techniques from ordinary differential equations, geometric analysis, orgeometric PDEs. Of particular interest are the results focused on minimal sub-manifolds and their connection with various geometric functionals. Additionally,geometers have actively studied other classes of geometric objects such as totallyumbilical submanifolds, ideal immersions, Lagrangian submanifolds, complex andtotally real submanifolds, and submanifolds of finite type. Our research interestsinclude the study of curvature functionals in various contexts and ambient spaces,comparison geometry, geometric PDEs, relations between curvature and topology,and other related topics. The works included in the present volume illustrate manyof these ideas.

The present volume includes papers presented in two AMS Special Sessions.The first event was the AMS Special Session on Geometry of Submanifolds, whichtook place on October 25–26, 2014, at San Francisco State University, during theWestern Fall Sectional Meeting (Meeting #1104). The second event was the AMSSpecial Session on Recent Advances on Submanifold Geometry, Dedicated to theMemory of Franki Dillen (1963–2013), which took place on March 14–15, 2015,East Lansing, during the Spring Central Sectional Meeting (Meeting #1108). We

vii

viii PREFACE

extend our thanks to David Bao, Chair of the Department of Mathematics at SanFrancisco State University, and to Keith Promislow, Chair of the Department ofMathematics at Michigan State University, for all the efforts that they and theircollaborators invested in organizing the two conferences.

The reason that the second AMS Special Session was hosted in East Lansingis that Franki Dillen developed many research projects in collaboration with hisMichigan State University co-authors, namely Bang-Yen Chen and David E. Blair.Out of these collaborations new ideas flourished. They are still inspiring manyscholars working in the geometry of submanifolds. Of particular importance isthe discussion of the proof of the normal scalar curvature conjecture, a questionraised in 1999 by Franki Dillen and his collaborators and solved first by Zhiqin Lu,and then, independently, by Jianquan Ge and Zizhou Tang. Zhiqin Lu continuedhis investigation through his work with David Wenzel, which is included in thisvolume, along with a paper of I. Mihai investigating extensions of the same class ofinequalities.

Among the most important questions still open in the geometry of submani-folds, we should mention those conjectures attributable to Bang-Yen Chen. In 1991,he formulated the biharmonic conjecture. This claims that minimal submanifoldsare the only biharmonic submanifolds in Euclidean spaces. Additionally, in 1996he conjectured that every finite type spherical hypersurface is either of 1-type orof 2-type. He also conjectured that the only finite type closed hypersurfaces of aEuclidean space are the hyperspheres. For further details about these importantquestions, a recent comprehensive reference is Bang-Yen Chen’s monograph TotalMean Curvature and Submanifolds of Finite Type: 2nd Edition (World Scientific,2015). More details about these investigations are included in Ye-Lin Ou’s paperfrom the present volume.

A few years ago the geometry of submanifolds experienced a major develop-ment when Fernando Coda Marques and Andre Neves proved the classical WillmoreConjecture (originally asked in 1965). They use Almgren–Pitts min-max theory ofminimal surfaces to prove that the integral of the square of the mean curvature ofa torus in the three-dimensional Euclidean space is at least 2π2. It is natural tospeculate as to what new classes of problems researchers in the geometry of sub-manifolds will focus on in subsequent decades. Are there any important questionswhere the new techniques developed in the larger realm of contemporary differentialgeometry could make a major difference? By brainstorming on the fundamentalproblems and exploring a large variety of questions studied in submanifold geom-etry, the editors hope to provide mathematicians with a working tool, not just acollection of individual contributions.

The editors would like to extend their thanks to all the scholars who partici-pated in the two AMS Special Sessions. Their expertise and their interactions havebeen particularly valuable and interesting. While their papers are not includedin the present volume, the contributions of Ivko Dimitric (Penn State University),Weiyong He (University of Oregon), Martin Magid (Wellesley College), TommyMurphy (Cal State Fullerton), Mihaela Vajiac (Chapman University), Peng Wu(Cornell University), and Handan Yildirim (University of Istanbul) have been ex-tremely valuable to and tremendously appreciated by the editors. Also, manythanks to the co-authors of the contributors to the special sessions: Nikos Georgiou(Universidade de Sao Paulo), Martha Patricia Dussan Angulo (Universidade de

PREFACE ix

Sao Paulo), Changping Wang (Normal University of Fujian), and Jingyang Zhong(University of California, Santa Cruz).

The editors of the present volume express their thanks to Michel Lapidus andGeorgia Benkart, who served as AMS Secretaries in the academic year 2014–2015,when the two AMS Special Sessions were organized.

While the editors prepared the present volume, their work benefited from theoutstanding support and expert consultations of several referees. Without theirexpertise the quality of the present volume would not be the same. Last, but notleast, many thanks to Sergei Gelfand, Christine Thivierge, and Mike Saitas for theireditorial guidance and support during the preparation of the present volume.

The Editors

Selected Published Titles in This Series

674 Bogdan D. Suceava, Alfonso Carriazo, Yun Myung Oh, and Joeri Van derVeken, Editors, Recent Advances in the Geometry of Submanifolds, 2016

671 Robert S. Doran and Efton Park, Editors, Operator Algebras and TheirApplications, 2016

669 Sergiı Kolyada, Martin Moller, Pieter Moree, and Thomas Ward, Editors,Dynamics and Numbers, 2016

668 Gregory Budzban, Harry Randolph Hughes, and Henri Schurz, Editors,Probability on Algebraic and Geometric Structures, 2016

667 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 2: Complex Analysis, QuasiconformalMappings, Complex Dynamics, 2016

666 Vicentiu D. Radulescu, Adelia Sequeira, and Vsevolod A. Solonnikov, Editors,Recent Advances in Partial Differential Equations and Applications, 2016

665 Helge Glockner, Alain Escassut, and Khodr Shamseddine, Editors, Advances inNon-Archimedean Analysis, 2016

664 Dihua Jiang, Freydoon Shahidi, and David Soudry, Editors, Advances in theTheory of Automorphic Forms and Their L-functions, 2016

663 David Kohel and Igor Shparlinski, Editors, Frobenius Distributions: Lang-Trotterand Sato-Tate Conjectures, 2016

662 Zair Ibragimov, Norman Levenberg, Sergey Pinchuk, and Azimbay Sadullaev,Editors, Topics in Several Complex Variables, 2016

661 Douglas P. Hardin, Doron S. Lubinsky, and Brian Z. Simanek, Editors, ModernTrends in Constructive Function Theory, 2016

660 Habib Ammari, Yves Capdeboscq, Hyeonbae Kang, and Imbo Sim, Editors,Imaging, Multi-scale and High Contrast Partial Differential Equations, 2016

659 Boris S. Mordukhovich, Simeon Reich, and Alexander J. Zaslavski, Editors,Nonlinear Analysis and Optimization, 2016

658 Carlos M. da Fonseca, Dinh Van Huynh, Steve Kirkland, and Vu Kim Tuan,Editors, A Panorama of Mathematics: Pure and Applied, 2016

657 Noe Barcenas, Fernando Galaz-Garcıa, and Monica Moreno Rocha, Editors,Mexican Mathematicians Abroad, 2016

656 Jose A. de la Pena, J. Alfredo Lopez-Mimbela, Miguel Nakamura, and JimmyPetean, Editors, Mathematical Congress of the Americas, 2016

655 A. C. Cojocaru, C. David, and F. Pappalardi, Editors, SCHOLAR—a Scientific

Celebration Highlighting Open Lines of Arithmetic Research, 2015

654 Carlo Gasbarri, Steven Lu, Mike Roth, and Yuri Tschinkel, Editors, RationalPoints, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, 2015

653 Mark L. Agranovsky, Matania Ben-Artzi, Greg Galloway, Lavi Karp, DmitryKhavinson, Simeon Reich, Gilbert Weinstein, and Lawrence Zalcman, Editors,Complex Analysis and Dynamical Systems VI: Part 1: PDE, Differential Geometry, RadonTransform, 2015

652 Marina Avitabile, Jorg Feldvoss, and Thomas Weigel, Editors, Lie Algebras andRelated Topics, 2015

651 Anton Dzhamay, Kenichi Maruno, and Christopher M. Ormerod, Editors,Algebraic and Analytic Aspects of Integrable Systems and Painleve Equations, 2015

650 Jens G. Christensen, Susanna Dann, Azita Mayeli, and Gestur Olafsson,Editors, Trends in Harmonic Analysis and Its Applications, 2015

For a complete list of titles in this series, visit theAMS Bookstore at www.ams.org/bookstore/conmseries/.

This volume contains the proceedings of the AMS Special Session on Geometry of Sub-

manifolds, held from October 25–26, 2014, at San Francisco State University, San Fran-

cisco, CA, and the AMS Special Session on Recent Advances in the Geometry of Subman-

ifolds: Dedicated to the Memory of Franki Dillen (1963–2013), held from March 14–15,

2015, at Michigan State University, East Lansing, Ml.

The focus of the volume is on recent studies of submanifolds of Riemannian, semi-

Riemannian, Kaehlerian and contact manifolds. Some of these use techniques in classical

differential geometry, while others use methods from ordinary differential equations, geo-

metric analysis, or geometric PDEs. By brainstorming on the fundamental problems and

exploring a large variety of questions studied in submanifold geometry, the editors hope to

provide mathematicians with a working tool, not just a collection of individual contribu-

tions.

This volume is dedicated to the memory of Franki Dillen, whose work in submanifold

theory attracted the attention of and inspired many geometers.

ISBN978-1-4704-2298-1

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