We study harmonic maps from a 3-manifold with boundary to S 1 and prove a special case of dihedral rigidity of three dimensional cubes whose dihedral angles are π/2. Furthermore we give some applications to mapping torus hyperbolic 3-manifolds. Contents 1. Introduction 1 2. Formula for Laplacian of energy of harmonic map to the circle 3 3. Application to rigidity of scalar curvature and mean curvature 4 4. Application to hyperbolic mapping torus 10 4.1. Genus bound for mapping torus 11 4.2. Interpretation of φ and some applications 12 References 12
Using the upper half space model, we evaluate a component of the hyperbolic mass functional evaluated on a special family of polyhedra extending a formula of Miao-Piubello.
Built on a recent work of Almaraz, Barbosa, de Lima on positive mass theorems on asymptotically flat manifods with a noncompact boundary, we apply free boundary minimal surface techniques to prove their positive mass theorem and study the existence of positive scalar curvature metrics with mean convex boundary on a connected sum of the form (T n−1 × [0, 1])#M0.
We study hypersurfaces with prescribed null expansion in an initial data set. We propose a notion of stability and prove a topology theorem. Eichmair's Perron approach toward the existence of marginally outer trapped surface adapts to the settings of hypersurfaces with prescribed null expansion with only minor modifications.
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