This article finds constant scalar curvature Kähler metrics on certain compact complex surfaces. The surfaces considered are those admitting a holomorphic submersion to curve, with fibres of genus at least 2. The proof is via an adiabatic limit. An approximate solution is constructed out of the hyperbolic metrics on the fibres and a large multiple of a certain metric on the base. A parameter dependent inverse function theorem is then used to perturb the approximate solution to a genuine solution in the same cohomology class. The arguments also apply to certain higher dimensional fibred Kähler manifolds.

We prove that any cone having vertex density strictly between 1 and 3, occurring as the weak limit of a sequence of oriented, immersed, stable minimal hypersurfaces and lying near a pair of hyperplanes must itself be a pair of hyperplanes.

In this paper, we classify the stable properly embedded orientable minimal surfaces in M × ℝ, where M is a closed orientable Riemannian surface. We show that such a surface is a product of a stable embedded geodesic on M with ℝ, a minimal graph over a region of M bounded by stable geodesics, M × {t} for some t ∈ ℝ, or is in a moduli space of periodic multigraphs parametrized by P × ℝ⁺, where P is the set of primitive (non-multiple) homology classes in H₁(M).

We prove that the integral K-theoretic Novikov conjecture holds for torsion free arithmetic subgroups of linear algebraic groups.

In this paper, we prove that any two birational projective varieties with finite quotient singularities can be realized as two geometric GIT quotients of a non-singular projective variety by a reductive algebraic group. Then, by applying the theory of Variation of Geometric Invariant Theory Quotients ([3]), we show that they are related by a sequence of GIT wall-crossing flips.

We classify the tube domains in ℂ⁴ with affinely homogeneous base whose boundary contains a non-degenerate affinely homogeneous hypersurface. It follows that these domains are holomorphically homogeneous and amongst them, there are four new examples of unbounded homogeneous domains (that do not have bounded realisations). These domains lie to either side of a pair of Levi-indefinite hypersurfaces. Using the geometry of these two hypersurfaces, we find the automorphism groups of the domains.