We develop a general approach that uses holomorphic 1-forms to parameterize anatomical surfaces with complex (possibly branching) topology. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. Based on Riemann surface structure, we can then canonically partition the surface into patches. Each of these patches can be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. To illustrate the technique, we computed conformal structures for several types of anatomical surfaces in MRI scans of the brain, including the cortex, hippocampus, and lateral ventricles. We found that the resulting parameterizations were consistent across

We prove that every homomorphism $\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{O}^{F}_{\zeta}$ , with$E$and$F$Banach spaces and ζ∈ℂ^{$m$}, is induced by a $\mathop{\mathrm{Hom}}(E,F)$ -valued holomorphic germ, provided that 1≤$m$<∞. A similar structure theorem is obtained for the homomorphisms of type $\mathcal{O}^{E}_{\zeta}\rightarrow\mathcal{S}_{\zeta}$ , where $\mathcal{S}_{\zeta}$ is a stalk of a coherent sheaf of positive depth. We later extend these results to sheaf homomorphisms, obtaining a condition on coherent sheaves which guarantees the sheaf to be equipped with a unique analytic structure in the sense of Lempert–Patyi.

We associate to any given finite set of valuations on the polynomial ring in two variables over an algebraically closed field a numerical invariant whose positivity characterizes the case when the intersection of their valuation rings has maximal transcendence degree over the base fields. As an application, we give a criterion for when an analytic branch at infinity in the affine plane that is defined over a number field in a suitable sense is the branch of an algebraic curve.

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.

We discuss the uniruledness of various base loci of linear systems related to the canonical divisor. In particular we prove that the stable base locus of the canonical divisor of a smooth projective variety of general type is covered by rational curves.