Solution To The Normal Scalar Curvature Conjecture And Related Problems

The famous normal scalar curvature conjecture (also called DDVV conjecture)states that there is a pointwise inequality involving the scalar curvature, the normalscalar curvature and the mean curvature on a submanifold of a real space form. Sincethis conjecture has submanifold's intrinsic invariant and extrinsic invariant being in anoptimal inequality, it commendably shows the close relationship between the submani-fold and its ambient space and thus holds important position in submanifolds geometry.Incidentally, the algebraic version of this conjecture takes up great importance in thetheory of random matrices and its low rank cases also have significant applications inthe well-known scalar curvature pinching problem. In these years, many geometers aretrying to prove this conjecture and looking for ways to characterize and classify thosesubmanifolds on which the equality in the conjecture holds everywhere, but there'reonly some partial results for low rank cases. As one of the main results of this pa-per, using continuity method, we completely prove this conjecture together with thecondition for the equality holding at one point. In the same way, we get another opti-mal inequality similar to the algebraic version of this conjecture and also its equalitycondition。On the other hand, by using moving frame theory, we introduce"2p-th meancurvature"and"(2p + 1)-th mean curvature vector field"for a general submanifold.And then give an integral expression for them that characterizes them as mean values ofsymmetric functions of the principal curvatures of the submanifold. The normal scalarcurvature conjecture is thus a pointwise inequality involving"2nd mean curvature","1st mean curvature vector field"and the normal scalar curvature, which is a localproblem. Now we apply this integral expression formula to study global problems re-lated to these mean curvatures: (1) give a unified geometric proof of the well-knownnormal degree theorem and tangential degree theorem in which the first theorem wasfirstly proved by S.S.Chern with geometrical method and both were proved by Lashofand Smale with topological method; (2) give a simple and direct proof of the well-known Gray-Weyl tube's volume formula; (3) give a tubular proof of some Minkowski type integral formulas for high co-dimensional submanifolds; (4) discuss a special vari-ational problem, whose critical point is called tubular minimal submanifold, and studyits relation with austere submanifold.Lastly, we introduce r-th anisotropic mean curvature for hypersurfaces in Eu-clidean space and prove the anisotropic version of the Alexandrov-Reilly-Ros theo-rem, i.e., the Wulff shape is the only closed embedded hypersurface with constant r-thanisotropic mean curvature for some r, in which the case r = 1 gives an affirmativeanswer to an open problem of F.Morgan.