Sinica-NCTS Geometry Seminar
主講者: Professor Gabor Toth (Rutgers University, Camden)
講題: Moduli of minimal orbits in spheres
時間: 2014-01-17 (Fri.)  16:00 - 17:00
地點: 數學所 617 研討室 (台大院區)
Abstract: Let $M=G/K$ be a compact Riemannian homogeneous manifold, $\lambda$ an eigenvalue of the Laplace-Beltrami operator $\triangle$ acting on $C^{\infty}(M)$, and $H_{\lambda}\subset C^{\infty}(M)$ the corresponding eigenspace. A spherical $\lambda$-eigenmap $f:M\to S_V$ into the unit sphere of a Euclidean vector space $V$ is characterized by having its components in $H_{\lambda}$. A $\lambda$-eigenmap is a harmonic map of constant energy density in the sense of Eells-Sampson. A conformal $\lambda$-eigenmap is called a spherical minimal immersion; it is an isometric minimal immersion of $M$ into $S_V$ with respect to $\lambda/\dim M$-times the original metric on $M$. For fixed $\lambda$, the set of all $\lambda$-eigenmaps can be parametrized by a compact convex body $L_{\lambda}$ in a finite dimensional $G$-module $E_{\lambda}$. Similarly, the set of all isometric minimal immersions can be parametrized by a compact convex body $M_{\lambda}$, a linear slice of $L_{\lambda}$ by a $G$-submodule $F_{\lambda}$$\subset$$E_{\lambda}$. A fundamental problem is to determine the highest weights of the irreducible $G$-components of $E_{\lambda}$ and $F_{\lambda}$ which thereby give the dimensions of $L_{\lambda}$ and $M_{\lambda}$ and rigidity. Beyond this quest one aims to understand the geometry of these moduli via Minkowski-type measures of symmetry developed for general convex sets. The aim of this talk is to define a sequence of such measures and calculate them in specific instances. This, in particular, will give a new understanding of the "roundness" of the space of minimal $SU(2)$-orbits in spheres.
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