Using the weak solution of Inverse mean curvature flow, we prove the sharp Minkowski-type inequality for outward minimizing hypersurfaces in Schwarzschild space.

The classification work [5], [8] left unsettled only those anomalous isoparametric hypersurfaces with four principal curvatures and multiplicity pair {4, 5}, {6, 9}, or {7, 8} in the sphere. By systematically exploring the ideal theory in commutative algebra in conjunction with the geometry of isoparametric hypersurfaces, we show that an isoparametric hypersurface with four principal curvatures and multiplicities {4, 5} in S19 is homogeneous, and, moreover, an isoparametric hypersurface with four principal curvatures and multiplicities {6, 9} in S31 is either the inhomogeneous one constructed by Ferus, Karcher, and M¨unzner, or the one that is homogeneous. This classification reveals the striking resemblance between these two rather different types of isoparametric hypersurfaces in the homogeneous category, even though the one with multiplicities {6, 9} is of the type constructed by Ferus, Karcher, and M¨unzner and the one with multiplicities {4, 5} stands alone. The quaternion and the octonion algebras play a fundamental role in their geometric structures. A unifying theme in [5], [8], and the present sequel to them is Serre’s criterion of normal varieties. Its technical side pertinent to our situation that we developed in [5], [8] and extend in this sequel is instrumental. The classification leaves only the case of multiplicity pair {7, 8} open.

No abstract uploaded!

This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embedding into the reference spacetime, the Killing fields of the reference spacetime are transplanted back to the surface in the physical spacetime to complete the definitions of quasi-local conserved quantities. We also compute how the corresponding total conserved quantities evolve under the Einstein equation with a cosmological constant.

For a toric Calabi-Yau (CY) orbifold X whose underlying toric variety is semi-projective, we construct and study a non-toric Lagrangian torus fibration on X, which we call the Gross fibration. We apply the Strominger-Yau-Zaslow (SYZ) recipe to the Gross fibration of X to construct its mirror with the instanton corrections coming from genus 0 open orbifold Gromov-Witten (GW) invariants, which are virtual counts of holomorphic orbi-disks in X bounded by fibers of the Gross fibration. We explicitly evaluate all these invariants by first proving an open/closed equality and then employing the toric mirror theorem for suitable toric (parital) compactifications of X. Our calculations are then applied to (1) prove a conjecture of Gross-Siebert on a relation between genus 0 open orbifold GW invariants and mirror maps of X – this is called the open mirror theorem, which leads to an enumerative meaning of mirror maps, and (2) demonstrate how open (orbifold) GW invariants for toric CY orbifolds change under toric crepant resolutions – an open analogue of Ruan’s crepant resolution conjecture.