Tues. December 20, 2011 9:45-10:00 　Registration 10:00-11:30 　Richard Schoen 11:45-12:45 　Shigeru Sakaguchi   14:00-15:00 　Takeyuki Nagasawa 15:15-16:15 　Eric Loubeau 16:30-17:30 　Keisuke Ueno Wed. December 21, 2011 9:45-11:15 　Qi Zhang 11:30-12:30 　Wayne Rossman    14:00-15:00 　Yasuhiro Nakagawa 15:15-16:15 　Sumio Yamada 16:30- 　Closing Richard Schoen (Stanford University) Tues. 10:00-11:30 "Eigenvalue problems and minimal surfaces in the ball and the sphere"

Abstract: We first survey some connections between extremal surfaces for eigenvalue problems and minimal surfaces. For surfaces with boundary we consider the spectrum of the Dirichlet-Neumann map. This is the spectrum of the operator which sends a function on the boundary of a manifold to the normal derivative of its harmonic extension. Along with the Dirichlet and Neumann spectrum, this eigenvalue problem has been classically studied. We show how the question of finding surfaces with fixed boundary length and largest first eigenvalue is connected to the study of minimal surfaces in the ball which meet the boundary orthogonally (free boundary solutions). We describe some conjectures and results on determining optimal surfaces. This is joint work with Ailana Fraser.

Shigeru Sakaguchi (Hiroshima University) Tues. 11:45-12:45 "Stationary isothermic surfaces and Liouville-type theorems characterizing hyperplanes"

Abstract: We consider a class W of Weingarten hypersurfaces in R^N with N > 1 , which contains an example related to stationary isothermic hypersurfaces. Denote by C the class of continuous entire graphs x_{N+1} = f(x), over R^N such that the oscillation of f restricted on each unit ball is bounded.Then, our main theorem is stated as follows: If a graph in the class C belongs to the class W in the viscosity sense, then S must be a hyperplane. This theorem gives considerable improvements of the previous results of S. Sakaguchi “A Liouville-type theorem for some Weingarten hypersurfaces”, Discrete and Continuous Dynamical Systems - Series S 4 (2011), 887-895 and R. Magnanini and S. Sakaguchi “Stationary isothermic surfaces and some characterizations of the hyperplane in the N-dimensional Euclidean space”, J. Differential Equations 248 (2010), 1112-1119.

Takeyuki Nagasawa (Saitama University) Tues. 14:00-15:00 "On the global existence of generalized rotational hypersurfaces with prescribed mean curvature in the Euclidean spaces "

Abstract: The history of construction of rotational surfaces with prescribed mean curvature began with Delaunay's study. After him, his results were generalized by many researchers. Here we discuss the global existence of generalized rotational hypersurfaces with prescribed mean curvature in the Euclidean spaces. It is known that there exist five types (say, Types I-V) of generalized rotational hypersurfaces. The existence of such hypersurfaces with prescribed mean curvature H was shown by Kenmotsu for Type I in 3-dimensional space when H is continuous; in higher dimensional case by Dorfmeister-Kenmotsu when H is a given analytic function. We prove the global existence of all types assuming that H is merely continuous. The results for Type I-II are obtained by joint-research with Kenmotsu.

Eric Loubeau (University of Brest) Tues. 15:15-16:15 "The harmonicity of vector fields and sections"

Abstract: The theory of harmonic maps has brought to light many fascinating interactions between geometry and analysis, but its universal setting and analytical nature have often made it almost impossible to construct concrete examples. I will try to present here recent works where this theory is applied to restricted but very geometrical situations, namely fibres bundles and their sections. The first part will concentrate on tangent bundles and vector fields, so all examples will be explicit, though a complete picture remains out of reach. At the other end of the spectrum, a similar approach can be followed for geometric structures, e.g. almost complex or almost contact, via their defining sections into adequate twistor bundles.

Keisuke Ueno (Yamagata University) Tues. 16:30-17:30 "Conformal variation of the total Q-curvature and the stability"

Abstract: Let M be a compact Riemannian manifold of the dimension n > 2. Given a Riemannian metric g on M, let Q_{g} be the Q-curvature of a Riemannian manifold (M, g), that is, where Ric_{g} and R_{g} are the Ricci curvature and the scalar curvature of (M, g), respectively, and \Delta_{g} denotes the Laplace-Beltrami operator with non-positive spectrum. We note that if (M, g) is an Einstein manifold, then Q-curvature of (M, g) is a constant. For each Riemannan metric g on M we define the total Q-curvature Q[g] of g by the integral of Q_{g} over M. Then, under volume preserving conformal variations, Professor Nishikawa obtained the first variation formula for Q[g], and proved that the Q-curvature of the critical metric is a constant. In particular, every Einstein metric is a critical point of Q[g]. He also obtained the second variational formula for Q[g], and showed the stability of the Einstein metric of positive scalar curvature. I will talk about the outline of the proof and concerning results.

Qi Zhang (University of California at Riverside) Wed. 9:45-11:15 " Extra regularity for harmonic functions on manifold without Ricci lower bound"

Abstract:We will present a recent volume non-inflating property for Ricci flow in general and for Kähler-Ricci flow in particular. As one application, we study harmonic functions in a class of compact manifolds arising from Kähler Ricci flow, for which there is no known lower bound on the Ricci curvature. We prove these functions have extra regularity.

Wayne Rossman (Kobe University) Wed. 11:30-12:30 "Discrete linear Weingarten surfaces"

Abstract: We define discrete linear Weingarten surfaces in 3-dimensional space forms, in analogy to the case of smooth linear Weingarten surfaces. Since the discrete surfaces have no canonical notion of differentiation, we must use alternate characterizations of smooth linear Weingarten surfaces to establish the analogy. For this purpose, Lie sphere geometry and the notion of Omega surfaces is useful. .

Yasuhiro Nakagawa (Kanazawa University) Wed. 14:00-15:00 "New examples of Sasaki-Einstein manifolds;"

Abstract:In this talk, I shall explain the result of our joint work with Toshiki Mabuchi(Osaka University): Stimulated by the existence result by Futaki, Ono and Wang for toric Sasaki-Einstein metrics, we have constructed new examples of Sasaki-Einstein metrics on S^1-bundles associated to canonical line bundles of CP(1)-bundles over Kähler-Einstein Fano manifolds, even though the Futaki's obstruction does not vanish. Here our examples include non-toric Sasaki-Einstein manifolds.

Sumio Yamada (Tohoku University) Wed. 15:15-16:15 "New Finsler structures on Teichmüller Spaces"

Abstract: The Teichmüller space of Riemann surfaces of genus g>1 can be regarded as an open convex subset in an ambient space, called Coxeter-Teichmüller complex, which is the Weil-Petersson geodesic completion of the Teichmüller space. The boundary points of the convex body consists of nodal surfaces, resulting from degenerations of conformal structures. On the other hand, there is a Finsler metric called Funk metric on convex bodies in Euclidean spaces. In this talk, we import the Euclidean convex geometry, in particular the Funk geometry, to the Weil-Petersson convex geometry of Teichmüller spaces, and present a comparison with the pre-existing geometries of Teichmüller spaces, including the geometry associated with the Thurston's earthquake deformation theory. Conference　Organizer Seiki Nishikawa（西川青季） @math.tohoku.ac.jp Sumio Yamada （山田澄生） @math.tohoku.ac.jp 科学研究費・基盤研究 (B) 20340009 （研究代表者：西川青季）による研究集会