In this paper we prove that any immersed stable capillary hypersurfaces in a ball in space forms are totally umbilical. %either a totally geodesic disk or a spherical cap. Our result also provides a proof of a conjecture proposed by Sternberg-Zumbrun in {\it J Reine Angew Math 503 (1998), 63--85}. We also prove a Heintze-Karcher-Ros type inequality for hypersurfaces with free boundary in a ball, which, together with the new Minkowski formula, yields a new proof of Alexandrov's Theorem for embedded CMC hypersurfaces in a ball with free boundary.

The onset of thermal convection in a rapidly rotating spherical shell is studied by linear stability analysis based on the fully compressible Navier-Stokes equations. Compressibility is quantified by the number of density scale heights N-rho, which measures the intensity of density stratification of the motionless, polytropic base state. The nearly adiabatic flow with polytropic index n = 1.499 < na = 1.5 is considered, where na is the adiabatic polytropic index. By investigating the stability of the base state with respect to the disturbance of specified wavenumber, the instability process is found to be sensitive to the Prandtl number Pr and to N-rho. For large Pr and small N-rho, the quasi-geostrophic columnar mode loses stability first; while for relatively small Pr a new quasi-geostrophic compressible mode is identified, which becomes unstable first under strong density stratification. The inertial mode can also occur first for relatively small Pr and a certain intensity of density stratification in the parameter range considered. Although the Rayleigh numbers Ra for the onsets of the quasi-geostrophic compressible mode and columnar mode are different by several orders of magnitude, we find that they follow very similar scaling laws with the Taylor number. The critical Ra for convection onset is found to be always positive, in contrast with previous results based on the widely used anelastic model that convection can occur at negative Ra. By evaluating the relative magnitude of the time derivative of density perturbation in the continuity equation, we show that the anelastic approximation in the present system cannot be applied in the small-Ra and large-N-rho regime.

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In this paper we study generalized Gorenstein Arf rings: a class of one-dimensional Cohen–Macaulay local Arf rings that is strictly contained in the class of Gorenstein rings. We obtain new characterizations and examples of Arf rings, and give applications of our argument to numerical semigroup rings and certain idealizations. In particular, we generalize a beautiful result of Barucci and Fröberg concerning Arf numerical semigroup rings.

In this paper, we study the question of which compact manifolds admit a metric with positive scalar curvature. Scalar curvature is perhaps the weakest invariant among all the well-known invariants constructed from the curvature tensor. It measures the deviation of the Riemannian volume of the geodesic ball from the euclidean volume of the geodesic ball. As a result, it does not tell us much of the behavior of the geodesics in the manifold. Therefore it was remarkable that in 1963, Lichnerowicz [i] was able to prove the theorem that on a compact spin manifold with positive scalar curvature, there is no harmonic spinor. Applying