Global analysis of curves and surfaces

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We studied types of Jacobi fields of constant mean curvature surfaces. If the rotational surface is critical in the sense of stability, the Jacobi fields are usually rotationally symmetric. However, we found also rotationally non-symmetric one. The existence of non-symmetric Jacobi fields is not surprising, but the existence on the critical surface means that even symmetric stable solution of a variational problem may splits into non-symmetric solutions.We proved that the hemi-sphere is the unique stable solutions of the free boundary problem of constant mean curvature surfaces with boundary in a plain. And, got a sufficient condition on the size of boundary and surface to be unique in the class of same boundary.We classified homogeneous Einstein metrics on certain homogeneous spaces. In the classification, simplification of the algebraic equation and the efficiency of the algorism of solving algebraic equations are essential.We prived that compact k-symmetric twister bundles over compact symmetric spaces are projected to compact symmetric spaces by primitive maps of finite type.We got new estimates from below of the spectrum of general infinite graphs. The estimate is sharp on many cases.We proved that global Sobolev-Bergman kernel can be analytically continued with respect to its Sobolev degree.We generalized the theorem of Omori-Yau concerning the growth of the volume on complete riemannian manifold and the maximum principle.We gave a sufficient condition to be strong solution of the weak solution of a boundary value problem with non constant rank of boundary matrix.