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

Peter Li Shing-Tung Yau

Differential Geometry mathscidoc:1912.43465

Inventiones mathematicae, 69, (2), 269-291, 1982.6
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 composition of qo with the conformal automorphisms of S". We let Vc (n, qg) be the maximum 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 demonstrate 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 RP z is 6n and the conformal area of the square torus is 27c 2. We believe that the computation of the conformal area for general surfaces will be very important 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] r proved that 5lk]> 4~ z if cr is
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@inproceedings{peter1982a,
  title={A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces},
  author={Peter Li, and Shing-Tung Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203433040378029},
  booktitle={Inventiones mathematicae},
  volume={69},
  number={2},
  pages={269-291},
  year={1982},
}
Peter Li, and Shing-Tung Yau. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. 1982. Vol. 69. In Inventiones mathematicae. pp.269-291. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191224203433040378029.
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