In this paper, we prove a general existence theorem for properly embedded minimal surfaces with free boundary in any compact Riemannian 3‐manifold M with boundary ∂M. These minimal surfaces are either disjoint from ∂M or meet ∂M orthogonally. The main feature of our result is that there is no assumptions on the curvature of M or convexity of ∂M. We prove the boundary regularity of the minimal surfaces at their free boundaries. Furthermore, we define a topological invariant, the filling genus, for compact 3‐manifolds with boundary and show that we can bound the genus of the minimal surface constructed above in terms of the filling genus of the ambient manifold M. Our proof employs a variant of the min‐max construction used by Colding and De Lellis on closed embedded minimal surfaces, which were first developed by Almgren and Pitts.

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to 1. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

In this paper we study the prescribed centroaffine curvature problem in the Euclidean space R n+ 1. This problem is equivalent to solving a Monge鈥揂mp猫re equation on the unit sphere. It corresponds to the critical case of the Blaschke鈥揝antal贸 inequality. By approximation from the subcritical case, and using an obstruction condition and a blow-up analysis, we obtain sufficient conditions for the a priori estimates, and the existence of solutions up to a Lagrange multiplier.

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For a random walk on a finitely generated group G we obtain a generalization of a classical inequality of Ancona. We deduce as a corollary that the identity map on G extends to a continuous equivariant surjection from the Martin boundary to the Floyd boundary, with preimages of conical points being singletons. This provides new results for Martin compactifications of relatively hyperbolic groups.

In this paper the existence of positive 2\pi -periodic solutions to the ordinary differential equation \begin{equation*} u^{\prime\prime}+u=\frac{f}{u^3} \ \textrm{ in } \mathbb{R} \end{equation*}is studied, where 2\pi is a positive 2\pi -periodic smooth function. By virtue of a new generalized Blaschke鈥揝antal贸 inequality, we obtain a new existence result of solutions.

By studying the Seiberg-Witten equations on end-periodic manifolds, we give an obstruction on the existence of positive scalar curvature metric on compact 4-manifolds with the same homology as S1×S3. This obstruction is given in terms of the relation between the Frøyshov invariant of the generator of H3(X;Z) with the 4-dimensional Casson invariant λSW(X) defined by Mrowka-Ruberman-Saveliev. Along the way, we develop a framework that can be useful in further study of the Seiberg-Witten theory on general end-periodic manifolds.

We introduce a new geometric flow, called the chord shortening flow, which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord not orthogonal to the boundary would shrink to a point in finite time under the flow.

In this article, we study the small sphere limit of the Wang-Yau quasi-local energy defined in [18,19]. Given a point p in a spacetime N , we consider a canonical family of surfaces approaching p along its future null cone and evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on solving an "optimal embedding equation" whose solutions represent critical points of the quasi-local energy. For a spacetime with matter fields, the scenario is similar to that of the large sphere limit found in [7]. Namely, there is a natural solution which is a local minimum, and the limit of its quasi-local energy recovers the stress-energy tensor at p . For a vacuum spacetime, the quasi-local energy vanishes to higher order and the solution of the optimal embedding equation is more complicated. Nevertheless, we are able to show that there exists a solution which is a local minimum and that the limit of its quasi-local energy is related to the Bel-Robinson tensor. Together with earlier work [7], this completes the consistency verification of the Wang-Yau quasi-local energy with all classical limits.

We show that Norbury’s McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of π1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

The skeleton of a 2D shape is an important geometric structure in pattern analysis and computer vision. In this paper we study the skeleton of a 2D shape in a two-manifold M, based on a geodesic metric. We present a formal definition of the skeleton S(\Omega) for a shape V in M and show several properties that make S(\Omega) distinct from its Euclidean counterpart in R^2. We further prove that for a shape sequence {\Omega_i} that converge to a shape V in M, the mapping \Omega\rightarrow S(\Omega_i) is lower semi-continuous. A direct application of this result is that we can use a set P of sample points to approximate the boundary of a 2D shape V in M, and the Voronoi diagram of P inside \Omega\subset M gives a good approximation to the skeleton S(\Omega). Examples of skeleton computation in topography and brain morphometry are illustrated.

In studying the "11/8-Conjecture" on the Geography Problem in 4-dimensional topology, Furuta proposed a question on the existence of Pin(2)-equivariant stable maps between certain representation spheres. In this paper, we present a complete solution to this problem by analyzing the Pin(2)-equivariant Mahowald invariants. As a geometric application of our result, we prove a "10/8+4"-Theorem. We prove our theorem by analyzing maps between certain finite spectra arising from BPin(2) and various Thom spectra associated with it. To analyze these maps, we use the technique of cell diagrams, known results on the stable homotopy groups of spheres, and the j-based Atiyah-Hirzebruch spectral sequence.

The purpose of this paper is to study the monodromy groups associated to the quasi-bounded holomorphic quadratic forms on punctured surfaces. As a consequence, we obtain a natural family of symplectic structures on the Teichm/iller space Tg,, for n> 0. As another consequence, we show that the projective monodromy map from a class of Fuchsian equations to the representation variety is generically a local diffeomorphism.

In this paper the centroaffine Minkowski problem, a critical case of the L p-Minkowski problem in the n+ 1 dimensional Euclidean space, is studied. By its variational structure and the method of blow-up analyses, we obtain two sufficient conditions for the existence of solutions, for a generalized rotationally symmetric case of the problem.

The face numbers of simplicial complexes without missing faces of dimension larger than$i$are studied. It is shown that among all such ($d$−1)-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the componentwise minimal$f$-vector; and moreover, among all such 2-Cohen–Macaulay (2-CM) complexes, the same sphere has the componentwise minimal$h$-vector. It is also verified that the$l$-skeleton of a flag ($d$−1)-dimensional 2-CM complex is 2($d$−$l$)-CM, while the$l$-skeleton of a flag piecewise linear ($d$−1)-sphere is 2($d$−$l$)-homotopy CM. In addition, tight lower bounds on the face numbers of 2-CM balanced complexes in terms of their dimension and the number of vertices are established.

In this paper, we prove uniform curvature estimates for immersed stable free boundary minimal hypersurfaces satisfying a uniform area bound, which generalize the celebrated Schoen–Simon–Yau interior curvature estimates up to the free boundary. Our curvature estimates imply a smooth compactness theorem which is an essential ingredient in the min-max theory of free boundary minimal hypersurfaces developed by the last two authors. We also prove a monotonicity formula for free boundary minimal submanifolds in Riemannian manifolds for any dimension and codimension. For 3-manifolds with boundary, we prove a stronger curvature estimate for properly embedded stable free boundary minimal surfaces without a-priori area bound. This generalizes Schoen’s interior curvature estimates to the free boundary setting. Our proof uses the theory of minimal laminations developed by Colding and Minicozzi.

In this paper, we solve a Monge-Ampre-type equation by the continuity method. This equation is motivated by the superstring theory.

The action of the mapping class group of the thrice-punctured projective plane on its GL(2,C) character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.

A surgery on a knot in 3-sphere is called SU(2)-cyclic if it gives a manifold whose fundamental group has no non-cyclic SU(2) representations. Using holonomy perturbations on the Chern-Simons functional, we prove that the distance of two SU(2)-cyclic surgery coefficients is bounded by the sum of the absolute values of their numerators. This is an analog of Culler-Gordon-Luecke-Shalen's cyclic surgery theorem.

For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

Here we classify$J$-embeddable surfaces, i.e. surfaces whose secant varieties have dimension at most 4, when the surfaces have two components at most.

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We establish a boundary maximum principle for free boundary minimal submanifolds in a Riemannian manifold with boundary, in any dimension and codimension. Our result holds more generally in the context of varifolds.

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Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology 3-spheres. As an application, we give explicit constrains on the intersection forms of spin 4-manifolds bounded by Brieskorn spheres ±Σ(2,3,6k±1). Along the way, we also give an alternative proof of Furuta-Kametanni's improvement of 10/8-theorem for closed spin-4 manifolds.