We study minimal graphic functions on complete Riemannian manifolds $\Sigma$ with non-negative Ricci curvature, Euclidean volume growth and quadratic curvature decay. We derive global bounds for the gradients for minimal graphic functions of linear growth only on one side. Then we can obtain a Liouville type theorem with such growth via splitting for tangent cones of $\Sigma$ at infinity. When, in contrast, we do not impose any growth restrictions for minimal graphic functions, we also obtain a Liouville type theorem under a certain non-radial Ricci curvature decay condition on $\Sigma$. In particular, the borderline for the Ricci curvature decay is sharp by our example in the last section.

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.

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Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle E over X. When pg(X) = 0, we construct deformations of holomorphic structures on E such that the new bundle is trivial over any ADE curve C inside C and therefore descends to the singular surface obtained by contracting C.

We carry out the programme of R. Liouville [19] to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure [..] on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of [..] or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

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Durch Doppelreihenentwicklung nach Legendreschen Polynomen wird die Lösung des Dirichletschen Problems mittels Picard-Iteration für einen beliebigen symmetrischen oder unsymmetrischen Rand numerisch durchgeführt. Für die erste Näherung der Lösung werden die Randwerte auf eine die Randkurve möglichst eng umschliessende Ellipse übertragen. Diese erste Näherung wird korrigiert durch Addition von Lösungen der homogen gemachten Differentialgleichung (Laplacesche Differentialgleichung), die den Fehler längs der Randkurve nach der Methode der kleinsten Quadrate zueinem Minimum machen. Die Berechnung dieser Korrektur wird besonders einfach im Falle eines achsensymmetrischen Randes. Ein Beispiel erläutert das Vorgehen.

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