This can be extended to a subset of cmc immersions of the plane which are of finite type. Sinestrari abstract we consider the evolution by mean curvature. Ruled surfaces of finite type bulletin of the australian. Pdf total mean curvature and submanifolds of finite type. The gap theorem is a generalized version of carron. Introduction the wellknown theorem of catalan states that the only ruled minimal sur. Inoguchi submanifolds with harmonic mean curvature vector field in contact 3manifolds colloq. Chen total mean curvature and submanifolds of finite type ser.
A gap theorem on submanifolds with finite total curvature in. Open problems and conjectures on submanifolds of finite. L 2 harmonic 1forms on submanifolds with finite total curvature springerlink. This cited by count includes citations to the following articles in scholar. The theory of total mean curvature is the study of the integral of the nth power of the mean curvature of a compact ndimensional submanifold. A list of ten open problems and three conjectures on submanifolds of finite type was published in 1981. If this summation is a finite sum, f is said to be of finite type. Buy total mean curvature and submanifolds of finite type.
Finite type submanifolds and generations, university of rome, italy, 1985. Total mean curvature and submanifolds of finite type series in pure mathematics book 27 kindle edition by bangyen chen. Enter your mobile number or email address below and well send you a link to download the free kindle app. Bangyen chen, total mean curvature and submanifolds of finite type, series in pure mathematics, vol. Also, hasanis and vlachos proved that biharmonic hypersurfaces in e 4.
This unique and expanded second edition comprises a comprehensive account of the latest updates and new results that cover total mean curvature and submanifolds of finite type. L 2 harmonic 1forms on submanifolds with finite total. Their combined citations are counted only for the first article. A new approach to compact symmetric spaces and applications a report on joint work with professor t. The tangential variation on hyperruled surfaces the tangential variation on hyperruled surfaces abdel all, n. Total mean curvature and submanifolds of finite type world scientific series in pure mathematicsvolume 1 by b ang. During the last four decades, there were numerous important developments on total mean curvature and the theory of finite type submanifolds. Use features like bookmarks, note taking and highlighting while reading total mean curvature and submanifolds of finite type series in pure mathematics book 27. Y total mean curvature and submanifolds of finite type, world scientific, singapore, 1984. Total mean curvature of surfaces in euclidean 3space 219 22. Assume that the immersion has finite total curvature in the sense that the traceless second fundamental form has finite l mnorm. Buy total mean curvature and submanifolds of finite type 2nd edition series in pure mathematics 2nd edition by bangyen chen isbn. L2 harmonic 1forms on complete submanifolds in euclidean space fu, haiping and li, zhenqi, kodai mathematical journal, 2009. Total mean curvature and submanifolds of finite type book.
The study of higher order energy functionals was first proposed by eells and sampson in 1965 and, later, by eells and lemaire in 1983. Total mean curvature and submanifolds of finite type, 2nd edition. Submission guidelines taiwanese journal of mathematics. In this paper, we consider surfaces in the 3dimensional euclidean space e3 which are of finite ii type, that is, they are of finite type, in the sense of b. Ruled submanifolds with finite type gauss map springerlink. Total mean curvature and submanifolds of finite type series. In fact, the main differential geometric ideas of the calculus of variation occur over and over again and are.
The well known chens conjecture on biharmonic submanifolds in euclidean spaces states that every biharmonic submanifold in a euclidean space is a minimal one. Sampson harmonic mappings of riemannian manifolds american j. Biharmonic submanifolds and biharmonic conjectures. The purpose of this book is to introduce the reader to two interesting topics in geometry which have developed over the last fifteen years, namely, total mean curvature and submanifolds of finite type. A gap theorem on submanifolds with finite total curvature. Total mean curvature and submanifolds of finite type series in. This unique and expanded second edition comprises a. We use cookies to distinguish you from other users and to provide you with a better experience on our websites. If m is embedded in r3, then we observe that m viewed from infinity looks like a plane passing through the origin. Furthermore u1 is composed of totally geodesic directions, and consequently we can reduce codimension to some e3 totally geodesic into em with. A new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces.
Biharmonic submanifolds of pseudoriemannian manifolds. Pdf during the last four decades, there were numerous important developments on total mean curvature and the theory of finite type. Mean curvature flows of lagrangian submanifolds with convex potentials knut smoczyk and mutao wang october 9, 2002 abstract this article studies the mean curvature ow of lagrangian submanifolds. Total mean curvature for arbitrary submanifolds and. Total mean curvature and submanifolds of finite type add library to favorites please choose whether or not you want other users to be able to see on your profile that this library is a favorite of yours.
This shows that the 2 type compact hypersurfaces of a hypersphere are masssymmetric. The theory of total mean curvature is the study of the integral of the nth power of the mean curvature of a compact ndimensional submanifold in a euclidean mspace and its applications to other branches of mathematics. Chen, total mean curvature and submanifolds of finite type, world. Mr 749575 5 bangyen chen, franki dillen, leopold verstraelen, and luc vrancken, ruled surfaces of finite type, bull. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in 40. Total mean curvature and submanifolds of finite type. Chen 6 in his program to understand finite types submanifolds, and. L2 harmonic 1forms on submanifolds with finite total curvature authors. Buy total mean curvature and submanifolds of finite type online of indias largest online book store, only genuine products. Total mean curvature and submanifolds of finite type by bang. The study of submanifolds of finite type began in the late. Download it once and read it on your kindle device, pc, phones or tablets.
Hitchin, pinkall, sterling and bobenko showed that all constant mean curvature immersions of a 2torus into the space forms, and can be described in purely algebrogeometric data. Submission guidelines please submit your manuscript as a pdf file here. In particular, we prove the following global existence and convergence theorem. For this case of stationary total curvature we prove a result on the quant,ization of the total curvature. L 2 harmonic 1forms on submanifolds with finite total curvature. Submanifolds with finite type gauss map volume 35 issue 2 bangyen chen, paolo piccinni.
Further results on total mean curvature for surfaces in euclidean space 220 22. Ruled submanifolds of finite type franki dillen communicated by jonathan m. Total mean curvature and submanifolds of finite type and millions of other books are available for amazon kindle. Ruled surfaces and tubes with finite type gauss map baikoussis, christos, chen, bangyen, and verstraelen, leopold, tokyo journal of mathematics, 1993. We present an important family of surfaces, namely, tubes in e3. Chen, total mean curvature and submanifolds of finite type, world scientific, singapore, 1984. Total mean curvature and submanifolds of finite type 2nd. The total curvature of riemannian manifolds and submanifolds has been a, field of ac,tive research during the last loro years. Total mean curvature and submanifolds of finite type by bangyen chen. Pdf on may 15, 1984, bangyen chen and others published total mean curvature and submanifolds of finite type find, read and cite all. Some classification results on finite type ruled submanifolds in a lorentzminkowski space kim, dongsoo and kim, young ho, taiwanese journal of mathematics, 2012. A list of twelve open problems and three conjectures on submanifolds of. These functionals provide a natural generalization of. May 03, 2007 an extrinsic rigidity theorem for submanifolds with parallel mean curvature in a sphere xu, hongwei, huang, fei, and xiang, fei, kodai mathematical journal, 2011.
M, m 3, be an isometric immersion of a complete noncompact manifold m in a complete simplyconnected manifold m with. We prove that the space of the l 2 harmonic 1forms on m has. Chens biharmonic conjecture and submanifolds with parallel. Mean curvature flows of lagrangian submanifolds with convex. A related series of results began with work of chern, do carmo and kobayashi cdck. If all coordinate functions of em restricted to m are of finite type, then m is said to be of finite type. For example, suppose that m is a complete minimal surface with finite total curvature. Submanifolds with finite type gauss map volume 35 issue 2 bangyen chen, paolo piccinni skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites.
Everyday low prices and free delivery on eligible orders. This means that v and so vl is a parallel subbundle in a. The theory of total mean curvature is the study of the integral of the nth power of the mean curvature of a compact ndimensional submanifold in a euclidean mspace and its applications to. Woodward curvature surfaces of submanifolds of pseudoriemannian manifolds with applications to complex space forms 31 r.
Cavalcante, heudson mirandola, feliciano vitorio submitted on 25 jan 2012 v1, last revised 5 jun 2012 this version, v2. Then you can start reading kindle books on your smartphone, tablet, or computer no kindle device required. We prove that there admit no nontrivial l 2harmonic 1forms on m if the total curvature is bounded from above by a constant depending only on n. The first results on this subject had been collected in authors book total mean curvature and sub manifolds of finite type, world scientific, nj, 1984. For hypersurfaces, we know from chen and jiang that the conjecture is true for biharmonic surfaces in e 3. L2 harmonic 1forms on submanifolds with finite total. The study of submanifolds of finite type began in 1970, with chens attempts to find the best possible estimate of the total mean curvature of a compact submanifold of the euclidean space and to find a notion of degree for euclidean submanifolds 1, 2. Total mean curvature of surfaces in euclidean 3space. On submanifolds with harmonic mean curvature 2547 that o must vanish identically. Some open problems and conjectures on submanifolds of finite. Some open problems and conjectures on submanifolds of finite type. Nagano, katholieke universiteit leuven, belgium, 1987.
Total mean curvature and submanifolds of finite type world. The tangential variation on hyperruled surfaces, applied. Ruled surfaces of finite type volume 42 issue 3 bangyen chen, franki dillen, leopold verstraelen, luc vrancken skip to main content we use cookies to distinguish you from other users and to provide you with a better experience on our websites. The main result of this paper is that if m is a compact flat surface in the euclidean mspace em, m. A gap theorem on submanifolds with finite total curvature in spheres. Woodward curvature surfaces of submanifolds of pseudoriemannian manifolds with applications to complex space forms 31. Total mean curvature and submanifolds of finite type by. Submanifolds with finite type gauss map bulletin of the. We show that a ruled submanifold with finite type gauss map in a euclidean space is a cylinder on a curve of finite type or a plane. Among other things, he derived the equations, in local coordinates, for spacelike biharmonic submanifolds in a pseudoriemannian manifold, and applied it to study spacelike biharmonic submanifolds of pseudoriemannian space forms with parallel mean curvature vector fields.
Ams proceedings of the american mathematical society. Total mean curvature and submanifolds of finite type, world scientific, 1984. The first results on this subject were collected in authors books 26,29. We show that a ruled submanifold of finite type in a euclidean space is a cylinder on a curve of finite type or a generalized helicoid. Submanifolds of finite type were introduced by the author during the late 1970s. Finite type submanifold, biharmonic submanifold, biharmonic. Buy total mean curvature and submanifolds of finite type books online at best prices in india by bangyen chen from. Ezenta minimal surfaces and the toda equations for the classical groups 22 j. Mean curvature flow of pinched submanifolds to spheres 3 a minimal surface with positive intrinsic curvature in a sphere, such as the veronese surface.
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