Singularities of plane complex curves and limits of K\"ahler metrics with cone singularities. I: Tangent Cones

We construct and classify, in the case of two complex dimensions, the possible tangent cones at points of limit spaces of non-collapsed sequences of K\"ahler-Einstein metrics with cone singularities.Comment: Reference to Panov's Polyhedral Kahler Manifolds adde

Asymptotically conical Ricci-flat Kahler metrics with cone singularities

The main result proved in this thesis is an existence theorem for asymptotically conical Ricci-flat Kahler metrics on C2 with cone singularities along a smooth complex curve. These metrics are expected to arise as blow-up limits of non-collapsed sequences of Kahler-Einstein metrics with cone singularities.Open Acces

Calabi-Yau metrics with cone singularities along intersecting complex lines: the unstable case

We produce local Calabi-Yau metrics on $\mathbf C^2$ with conical singularities along three or more complex lines through the origin whose cone angles strictly violate the Troyanov condition. The tangent cone at the origin is a flat polyhedral K\"ahler cone with conical singularities along two intersecting lines: one with cone angle corresponding to the line with smallest cone angle, while the other forms as the collision of the remaining lines into a single conical line. Using a branched covering argument, we can construct Calabi-Yau metrics with cone singularities along cuspidal curves with cone angle in the unstable range.Comment: Final version. Accepted in the Journal of the London Mathematical Societ

Some models for bubbling of (log) K\"ahler-Einstein metrics

We investigate aspects of the metric bubble tree for non-collapsing degenerations of (log) K\"ahler-Einstein metrics in complex dimensions one and two, and further describe a conjectural higher dimensional picture

Schauder estimates on products of cones

We prove an interior Schauder estimate for the Laplacian on metric products of two dimensional cones with a Euclidean factor, generalizing the work of Donaldson and reproving the Schauder estimate of Guo-Song. We characterize the space of homogeneous subquadratic harmonic functions on products of cones, and identify scales at which geodesic balls can be well approximated by balls centered at the apex of an appropriate model cone. We then locally approximate solutions by subquadratic harmonic functions at these scales to measure the H\"older continuity of second derivatives.Comment: 27 pages, 2 figure