Let M be a compact Riemannian manifold with a fixed conformal structure. Then we introduce the concept of conformal volume of M in the following manner. For each branched conformal immersion q9 of M into the unit sphere S n, we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of S". We let Vc(n, qg) be the max imum volume of these branched immersions. The conformal volume of M is defined to be the infimum of V.(n, q0) where qo ranges over all branched conformal immersions of M into the unit sphere S". In this paper, we study the case when M is a compact surface and we call the conformal volume of M to be the conformal area of M. We demonst ra te that this conformal invariant is non-trivial. In fact, we prove that if there exists a minimal immersion of M into S" where coordinate functions are first eigenfunctions, then the conformal area of M is given by the area of M with respect to the induced metric. This enables us to compute the conformal area for several surfaces. For example, the conformal area of R P z is 6n and the conformal area of the square torus is 27c 2. We believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces. We demonstrate this claim by applying the concept of conformal area to two different branches of surface theory. The first application is to study the total curvature of a compact surface in R". This problem has a long history. Fenchel and Fary [9] proved that for a closed curve o in R", Slk[>27c where k is its curvature. Then Milnor [12]

/pdf/a-new-conformal-invariant-and-its-applications-to-the-31lp62ubkr.pdf

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1978, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/a-general-comparison-theorem-with-applications-to-volume-2wcc88ic3f.pdf

A general comparison theorem with applications to volume estimates for submanifolds

The Willmore conjecture, proposed in 1965, concerns the quest to find the best torus of all. This problem has inspired a lot of mathematics over the years, helping bringing together ideas from subjects like conformal geometry, partial differential equations, algebraic geometry and geometric measure theory. In this article we survey the history of the conjecture and our recent solution through the min-max approach. We finish with a discussion of some of the many open questions that remain in the field.

The Willmore Conjecture

In [3], Cheng, Li and Yau proved comparison theorems (upper bounds) for the heat kernels on minimal submanifolds of space forms. In the present note we show that these comparison theorems together with a series of corollaries remain true for minimal submanifolds in ambient spaces with just an upper bound on the sectional curvature.

/pdf/on-the-heat-kernel-comparison-theorems-for-minimal-1uqvlhf935.pdf

On the heat kernel comparison theorems for minimal submanifolds

On a theorem of Fenchel-Borsuk-Willmore-Chern-Lashof

On the Total Curvature of Immersed Manifolds, III: Surfaces in Euclidean 4-Space

an (n -fp) -dimensional Eiemannian manifold Nn+p. Let h be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping T^XTx-* Tw-L for x ? M, where T^ is the tangent space of M at x and IVthe normal space to M at x. We let 8 denote the length of h, H the mean curvature vector of this immersion, and a the length of H. If a = 0 identically on M, then M is called a minimal submanifold of Nn+p. In the first paper of this series [1], the author proved that the integral of an satisfies

http://www.jstor.org/stable/pdfplus/2373759.pdf

On the Total Curvature of Immersed Manifolds, II: Mean Curvature and Length of Second Fundamental Form

where dV denotes the volume element of M2. The equality holds when and only when M2 is a sphere in E3. The result (A) was proved by Fenchel [5] in 1929, and (B) by Willmore [6] in 1968. The results (A) and (B) were generalized to closed curves and surfaces in higher dimensional e-Lelidean space by Borsuk [1] and Chen [3], respectively. In this paper we give some generalizations of (A) and (B). We consider a closed manifold Mn of dimension n and an immersion x: Mn > En+N of Mn into euclidean space of dimension n + N. Let B, (x) be the bundle of unit normal vectors of x (Mn) so that a point of B, (x) is a pair (p,e) where e is a unit normal vector at x(p). Then B,(x) is a bundle of (N -1)-dimensional spheres over Mn and is a (smooth) manifold of dimension n + N1. Let dV be the volume element of Mn. There is a differential form doof degree N 1 on B, (x) such that its restriction to a fibre is the volume element

On the total curvature of immersed manifolds, i: an inequality of fenchel-borsuk-willmore.

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On the Total Curvature of Immersed Manifolds, III: Surfaces in Euclidean 4-Space

On the Total Curvature of Immersed Manifolds, II: Mean Curvature and Length of Second Fundamental Form

On the total curvature of immersed manifolds, i: an inequality of fenchel-borsuk-willmore.