We study the simultaneous filling and embedding problem for a CR family of compact strongly pseudoconvex CR manifolds of dimension at least 5. We also derive, as a consequence, the normality of the Stein fibers of the filled-in Stein space under the constant dimensionality assumption of the first Kohn-Rossi cohomology group of the fiber CR manifolds. Two main ingredients for our approach are the work of Catlin on the solution of the ∂-equation with mixed boundary conditions and the work of Siu and Ling on the study of the Grauert direct image theory for a (1,1)-convex-concave family of complex spaces.

Motivated by the vacuum selection problem of string/M theory, we study a new geometric invariant of a positive Hermitian line bundle (L, h) → M over a compact Kähler manifold: the expected distribution of critical points of a Gaussian random holomorphic section s ∈ H⁰(M,L) with respect to the Chern connection ∇_h. It is a measure on M whose total mass is the average number N^crit_h of critical points of a random holomorphic section. We are interested in the metric dependence of N^crit_h, especially metrics h which minimize N^crit_h. We concentrate on the asymptotic minimization problem for the sequence of tensor powers (L^N, h^N) → M of the line bundle and their critical point densities K^crit_N,h(z). We prove that K^crit_N,h(z) has a complete asymptotic expansion in N whose coefficients are curvature invariants of h. The first two terms in the expansion of N^crit_N,h are topological invariants of (L,M). The third term is a topological invariant plus a constant β₂(m) (depending only on the dimension m of M) times the Calabi functional ∫_Mρ²dVol_h, where ρ is the scalar curvature of the Kähler metric ω_h := (i/2)Θ_h. We give an integral formula for β₂(m) and show, by a computer assisted calculation, that β₂(m) > 0 for m ≤ 5, hence that N^crit_N,h is asymptotically minimized by the Calabi extremal metric (when one exists). We conjecture that β₂(m) > 0 in all dimensions, i.e., the Calabi extremal metric is always the asymptotic minimizer.

We prove that polarised manifolds that admit a constant scalar curvature Kähler (cscK) metric satisfy a condition we call slope semistability. That is, we define the slope μ for a projective manifold and for each of its subschemes, and show that if X is cscK then μ(Z) ≤ μ(X) for all subschemes Z. This gives many examples of manifolds with Kähler classes which do not admit cscK metrics, such as del Pezzo surfaces and projective bundles. If P(E) → B is a projective bundle which admits a cscK metric in a rational Kähler class with sufficiently small fibres, then E is a slope semistable bundle (and B is a slope semistable polarised manifold). The same is true for all rational Kähler classes if the base B is a curve. We also show that the slope inequality holds automatically for smooth curves, canonically polarised and Calabi-Yau manifolds, and manifolds with c₁(X) > 0 and L close to the canonical polarisation.

Following Riemann's idea, we prove the existence of a minimal disk in Euclidean space bounded by three lines in generic position and with three helicoidal ends of angles less than π. In the case of general angles, we prove that there exist at most four such minimal disks, give a sufficient condition of existence in terms of a system of three equations of degree 2, and give explicit formulas for the Weierstrass data in terms of hypergeometric functions. Finally, we construct constant mean curvature one trinoids in hyperbolic space by the method of the conjugate cousin immersion.

S. Alesker has shown that if G is a compact subgroup of O(n) acting transitively on the unit sphere S^n-1, then the vector space Val^G of continuous, translation-invariant, G-invariant convex valuations on Rⁿ has the structure of a finite dimensional graded algebra over R satisfying Poincaré duality. We show that the kinematic formulas for G are determined by the product pairing. Using this result we then show that the algebra Val^U(n) is isomorphic to R[s, t]/(f_n+1, f_n+2), where s, t have degrees 2 and 1 respectively, and the polynomial f_i is the degree i term of the power series log(1 + s + t).