This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott’s original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperk¨ahler Kirwan map is surjective for the non-fixed determinant case.

We prove that the algebra of singular cochains on a smooth manifold, equipped with the cup product, is equivalent to the A1 structure on the Lagrangian Floer cochain group associated to the zero section in the cotangent bundle. More generally, given embeddings with isomorphic normal bundles of a closed manifold B into manifolds Q1 and Q2, we construct a differential graded category from the singular cochains of these spaces, and prove that it is equivalent to the A1 category obtained by considering exact Lagrangian embeddings of Q1 and Q2 which intersect cleanly along B.

In a series of works, one of the authors has developed with J.- M. Hwang a geometric theory of uniruled projective manifolds, especially those of Picard number 1, basing on the study of varieties of minimal rational tangents. A fundamental result in this theory is a principle of analytic continuation under very mild assumptions,called Cartan-Fubini extension, of biholomorphisms between connected open subsets of two Fano manifolds of Picard number 1 which preserve varieties of minimal rational tangents. In this article we develop a generalization of Cartan-Fubini extension for non-equidimensional holomorphic immersions from a connected open subset of a Fano manifold of Picard number 1 into a uniruled projective manifold, under the assumptions that the map sends varieties of minimal rational tangents onto linear sections of varieties of minimal rational tangents and that it satisfies a mild geometric condition formulated in terms of second fundamental forms on varieties of minimal rational tangents. Formerly such a result was known only in the very special case of irreducible Hermitian symmetric manifolds of rank at least two, and the proof relied on the existence of flattening coordinates, viz.,Harish-Chandra coordinates, with respect to which the varieties of minimal rational tangents form a constant family. The proof of the main result, which is based on the deformation theory of rational curves, is differential-geometric in nature and is applicable to the general situation of uniruled projective manifolds without any assumption on the existence of special coordinate systems. As an application, we give a characterization of standard embeddings for certain pairs of rational homogeneous manifolds in terms of embeddings of varieties of minimal rational tangents.

In 2004, Sormani and Wei introduced the covering spectrum: a geometric invariant that isolates part of the length spectrum of a Riemannian manifold. In their paper they observed that certain Sunada isospectral manifolds share the same covering spectrum, thus raising the question of whether the covering spectrum is a spectral invariant. In the present paper we describe a group theoretic condition under which Sunada’s method gives manifolds with identical covering spectra. When the group theoretic condition of our method is not met, we are able to construct Sunada isospectral manifolds with distinct covering spectra in dimension 3 and higher. Hence, the covering spectrum is not a spectral invariant. The main geometric ingredient of the proof has an interpretation as the minimum-marked-length-spectrum analogue of Colin de Verdi`ere’s classical result on constructing metrics where the first k eigenvalues of the Laplace spectrum have been prescribed.

Let (M, !) be a compact symplectic manifold, and L a compact embedded Lagrangian submanifold in M. Fukaya, Oh, Ohta and Ono [8] construct Lagrangian Floer cohomology, yielding groups HF(L, b;nov) for one or HF􀀀 (L1, b1), (L2, b2);nov
for two Lagrangians, where b, b1, b2 are choices of bounding cochains, and exist if and only if L,L1,L2 have unobstructed Floer cohomology. These are independent of choices up to isomorphism, and have important invariance properties under Hamiltonian equivalence. Floer cohomology groups are the morphism groups in the derived Fukaya category of (M, !), and so are an essential part of the Homological Mirror Symmetry Conjecture of Kontsevich. The goal of this paper is to extend [8] to immersed Lagrangians  : L ! M, with transverse self-intersections. In the embedded case, Floer cohomology HF(L, b;nov) is a modified, ‘quantized’ version of singular homology Hn−(L;nov) over the Novikov ring nov. In our immersed case, HF(L, b;nov) turns out to be a quantized version of Hn−(L;nov)  L (p−,p+)2R nov · (p−, p+), where R =  (p−, p+) : p−, p+ 2 L, p− 6= p+, (p−) = (p+) is a set of two extra generators for each self-intersection point of L, and (p−, p+) has degree (p−,p+) 2 Z, an index depending on how L intersects itself at (p−) = (p+). The theory becomes simpler and more powerful for graded Lagrangians in Calabi–Yau manifolds, when we can work over a smaller Novikov ring CY. The proofs involve associating a gapped filtered A1 algebra over 0 nov or 0 CY to  : L ! M, which is independent of nearly all choices up to canonical homotopy equivalence, and is built using a series of finite approximations called AN,0 algebras for N = 0, 1, 2, . . ..

We consider Kobayashi geodesics in the moduli space of abelian varieties Ag, that is, algebraic curves that are totally geodesic submanifolds for the Kobayashi metric. We show that Kobayashi geodesics can be characterized as those curves whose logarithmic tangent bundle splits as a subbundle of the logarithmic tangent bundle of Ag. Both Shimura curves and Teichm¨uller curves are examples of Kobayashi geodesics, but there are other examples. We show moreover that non-compact Kobayashi geodesics always map to the locus of real multiplication and that the Q-irreducibility of the induced variation of Hodge structures implies that they are defined over a number field.

We give a natural filtration F on QH(G/B), which respects the quantum product structure. Its associated graded algebra GrF(QH(G/B)) is isomorphic to the tensor product of QH(G/P) and a corresponding graded algebra of QH(P/B) after localization. When the quantum parameter goes to zero, this specializes to the filtration on H(G/B) from the Leray spectral sequence associated to the fibration P/B ! G/B −! G/P.

We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n  10, is highly nonconnected.

Let V be a complex nonsingular projective 3-fold of general type. We shall give a detailed classification up to baskets of singularities on a minimal model of V . We show that the m-canonical map of V is birational for all m≥73 and that the canonical volume Vol(V ) ≥ 1/2660 . When (OV ) ≤1, our result is Vol(V )≥1/420 , which is optimal. Other effective results are also included in the paper.

A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let  be an orthogonal subset of a set of simple roots associated with the symmetric space M. Then  determines a horospherical decomposition M = Fs ×ErankM−||×N, where Fs  is the Riemannian product of || symmetric spaces of rank one. Every hyperpolar homogeneous foliation on M is isometrically congruent to the product of the following objects: a particular homogeneous codimension one foliation on each symmetric space of rank one in Fs , a foliation by parallel affine subspaces on the Euclidean space ErankM−||, and the horocycle subgroup N of the parabolic subgroup of the isometry group of M determined by .

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We prove that energy minimizing Yang–Mills connections on compact homogeneous 4-manifolds are either instantons or split into a sum of instantons on passage to the adjoint bundle. We prove related results for Calabi–Yau 3-folds and for 3−dimensional manifolds of nonnegative Ricci curvature.

All continuous GL(n) covariant valuations that map convex bodies to convex bodies are completely classified. This establishes a characterization of moment and projection operators and shows that the Holmes-Thompson area is the unique Minkowski area that is also a bivaluation.

We prove that a constrained Willmore immersion of a 2–torus into the conformal 4–sphere S4 is of “finite type”, that is, has a spectral curve of finite genus, or of “holomorphic type” which means that it is super conformal or Euclidean minimal with planar ends in R4 = S4\{1} for some point 1 2 S4 at infinity. This implies that all constrained Willmore tori in S4 can be constructed rather explicitly by methods of complex algebraic geometry. The proof uses quaternionic holomorphic geometry in combination with integrable systems methods similar to those of Hitchin’s approach [19] to the study of harmonic tori in S3.

In this paper, we sketch the proof of the extension of the stability theorem of the Minkowski space in General Relativity done explicitly in [6] and [7]. We discuss solutions of the Einstein vacuum (EV) equations (obtained in the author’s Ph.D. thesis ([6]) in 2007). We solve the Cauchy problem for more general, asymptotically flat initial data than in the pioneering work of D. Christodoulou and S. Klainerman [21] or than in any other work. Moreover, we describe precisely the asymptotic behavior. Our relaxed assumptions on the initial data yield a spacetime curvature which is not bounded in L1(M). As a major result, we encounter in our work borderline cases, which we discuss in this paper as well. The fact that certain of our estimates are borderline in view of decay indicates that the conditions in our main theorem are sharp in so far as the assumptions on the decay at infinity on the initial data are concerned. Thus, the borderline cases are a consequence of our relaxed assumptions on the data, [6, 7]. They are not present in the other works, as all of them place stronger assumptions on their data. We work with an invariant formulation of the EV equations. Our main proof is based on a bootstrap argument. To close the argument, we have to show that the spacetime curvature and the corresponding geometrical quantities have the required decay. In order to do so, the Einstein equations are decomposed with respect to specific foliations of the spacetime. This result generalizes the work of D. Christodoulou and S. Klainerman [21].

Let G be a torsion-free hyperbolic group and let n  6 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n − 1)-dimensional sphere.

Based on large N Chern-Simons/topological string duality, in a series of papers [36, 21, 19], Labastida, Marino, Ooguri, and Vafa conjectured certain remarkable new algebraic structure of link invariants and the existence of infinite series of new integer invariants. In this paper, we provide a proof of this conjecture. Moreover, we also show these new integer invariants vanish at large genera.

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We construct a Kruskal-Szekeres-type analytic extension of the Emparan-Reall black ring, and investigate its geometry. We prove that the extension is maximal, globally hyperbolic, and unique within a natural class of extensions. The key to those results is the proof that causal geodesics are either complete, or approach a singular boundary in finite affine time. Alternative maximal analytic extensions are also constructed.

We study the algebraic dimension a(X) of a compact hyperk ¨ahler manifold of dimension 2n. We show that a(X) is at most n unless X is projective. If a compact K¨ahler manifold with algebraic dimension 0 and Kodaira dimension 0 has a minimal model, then only the values 0, n and 2n are possible. In case of middle dimension, the algebraic reduction is holomorphic Lagrangian. If n = 2, then - without any assumptions - the algebraic dimension only takes the values 0, 2 and 4. The paper also gives structure results for ”generalised hyperk¨ahler” manifolds and studies nef lines bundles.

We consider compact submanifolds of dimension n  2 in Rn+k, with nonzero mean curvature vector everywhere, and such that the full norm of the second fundamental form is bounded by a fixed multiple (depending on n) of the length of the mean curvature vector at every point. We prove that the mean curvature flow deforms such a submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in some (n + 1)- dimensional affine subspace of Rn+k.

We study the Plateau Problem of finding an area minimizing disk bounding a given Jordan curve in Alexandrov spaces with curvature ≥ κ. These are complete metric spaces with a lower curvature bound given in terms of triangle comparison. Imposing an additional condition that is satisfied by all Alexandrov spaces according to a conjecture of Perel’man, we develop a harmonic map theory from two dimensional domains into these spaces. In particular, we show that the solution to the Dirichlet problem from a disk is H¨older continuous in the interior and continuous up to the boundary. Using this theory, we solve the Plateau Problem in this setting generalizing classical results in Euclidean space (due to J. Douglas and T. Rado) and in Riemannian manifolds (due to C.B. Morrey).

Given a negatively curved geodesic metric space M, we study the almost sure asymptotic penetration behavior of (locally) geodesic lines of M into small neighborhoods of points, of closed geodesics, and of other compact (locally) convex subsets of M. We prove Khintchine-type and logarithm law-type results for the spiraling of geodesic lines around these objets. As a consequence in the tree setting, we obtain Diophantine approximation results of elements of non-archimedian local fields by quadratic irrational ones.

We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension d for all d ≥ 4 (see Theorem 2.13). These examples result from applying CAT(0)/CAT(−1) filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory (see Section 5).

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