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Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the ONeill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kahler-Einstein manifold or a Riemannian product of a Kahler-Einstein manifold with R (or with the circle S^1).
In this work we consider viscosity solutions to second order partial differential equations on Riemannian manifolds. We prove maximum principles for solutions to Dirichlet problem on a compact Riemannian manifold with boundary. Using a different meth
Let $(M,g)$ be a complete three dimensional Riemannian manifold with boundary $partial M$. Given smooth functions $K(x)>0$ and $c(x)$ defined on $M$ and $partial M$, respectively, it is natural to ask whether there exist metrics conformal to $g$ so t
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete
We study the index of the APS boundary value problem for a strongly Callias-type operator $D$ on a complete even dimensional Riemannian manifold $M$ (the odd dimensional case was considered in our previous paper arXiv:1706.06737). We use this index t
We generalize a Bernstein-type result due to Albujer and Alias, for maximal surfaces in a curved Lorentzian product 3-manifold of the form $Sigma_1times mathbb{R}$, to higher dimension and codimension. We consider $M$ a complete spacelike graphic sub