In this survey, we discuss some recent results on free boundary minimal surfaces in the Euclidean unit-ball. The subject has been a very active field of research in the past few years due to the seminal work of Fraser and Schoen on the extremal Steklov eigenvalue problem. We review several different techniques of constructing examples of embedded free boundary minimal surfaces in the unit ball. Next, we discuss some uniqueness results for free boundary minimal disks and the conjecture about the uniqueness of critical catenoid. We also discuss several Morse index estimates for free boundary minimal surfaces. Moreover, we describe estimates for the first Steklov eigenvalue on such free boundary minimal surfaces and various smooth compactness results. Finally, we mention some sharp area bounds for free boundary minimal submanifolds and related questions.
For any smooth Riemannian metric on an (n+1)-dimensional compact manifold with boundary (M,∂M) where 3≤(n+1)≤7, we establish general upper bounds for the Morse index of free boundary minimal hypersurfaces produced by min-max theory in the Almgren-Pitts setting. We apply our Morse index estimates to prove that for almost every (in the C-infinity Baire sense) Riemannan metric, the union of all compact, properly embedded free boundary minimal hypersurfaces is dense in M. If ∂M is further assumed to have a strictly mean convex point, we show the existence of infinitely many compact, properly embedded free boundary minimal hypersurfaces whose boundaries are non-empty. Our results prove a conjecture of Yau for generic metrics in the free boundary setting.
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.
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 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.
We prove a lower bound for the first Steklov eigenvalue of embedded minimal hypersurfaces with free boundary in a compact n-dimensional Riemannian manifold which has nonnegative Ricci curvature and strictly convex boundary. When n=3, this implies an apriori curvature estimate for these minimal surfaces in terms of the geometry of the ambient manifold and the topology of the minimal surface. An important consequence of the estimate is a smooth compactness theorem for embedded minimal surfaces with free boundary when the topological type of these minimal surfaces is fixed.
We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from , we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith . We also show that derived Fukaya categories are complete invariants of special isogenous tori.
In this paper, we prove the DonohoStark uncertainty principle for locally compact quantum groups and characterize the minimizer which are bi-shifts of group-like projections. We also prove the HirschmanBeckner uncertainty principle for compact quantum groups and discrete quantum groups. Furthermore, we show Hardy's uncertainty principle for locally compact quantum groups in terms of bi-shifts of group-like projections.
The representation category of a conformal net is a unitary modular tensor category. We investigate the reconstruction program: whether all unitary modular tensor categories are representation categories of conformal nets. We give positive evidence: the fruitful theory of multi-interval Jones-Wassermann subfactors on conformal nets is also true for modular tensor categories. We construct multi-interval Jones-Wassermann subfactors for unitary modular tensor categories. We prove that these subfactors are symmetrically self-dual. It generalizes and categorifies the self-duality of finite abelian groups. We call this duality the modular self-duality, because the modularity of the modular tensor category appears in a crucial way. For each unitary modular tensor category, we obtain a sequence of unitary fusion categories. The cyclic group case gives examples of Tambara-Yamagami categories.
In this article, we classify all standard invariants that can arise from a composed inclusion of an A 3 with an A 4 subfactor. More precisely, if N P is an A 3 subfactor and P M is an A 4 subfactor, then only four standard invariants can arise from the composed inclusion N M. We answer a question posed by Bisch and Haagerup in 1994. The techniques of this paper also show that there are exactly four standard invariants for the composed inclusion of two A 4 subfactors.
The classical uncertainty principles deal with functions on abelian groups. In this paper, we discuss the uncertainty principles for finite index subfactors which include the cases for finite groups and finite dimensional Kac algebras. We prove the HausdorffYoung inequality, Young's inequality, the HirschmanBeckner uncertainty principle, the DonohoStark uncertainty principle. We characterize the minimizers of the uncertainty principles and then we prove Hardy's uncertainty principle by using minimizers. We also prove that the minimizer is uniquely determined by the supports of itself and its Fourier transform. The proofs take the advantage of the analytic and the categorial perspectives of subfactor planar algebras. Our method to prove the uncertainty principles also works for more general cases, such as Popa's <i></i>-lattices, modular tensor categories, etc.
In this paper we generalize a result in [J. An, Z. Wang, On the realization of Riemannian symmetric spaces in Lie groups, Topology Appl. 153 (7) (2005) 10081015, showing that an arbitrary Riemannian symmetric space can be realized as a closed submanifold of a covering group of the Lie group defining the symmetric space. Some properties of the subgroups of fixed points of involutions are also proved.
With the rapid development of internet economy, transparent logistics is stepping into a prosperity period with massive transportation data generated and collected every day. In this paper, we focus on the segmentation of GPS trajectory data generated in logistics transportation to analyze the vehicle behaviors and extract business affair information according to the vehicle behavior characteristics, which is challenging due to the complexity of trajectory data and unavailability of road information. We extract the stopping points from the trajectory data sequence based on the duration of nonmovement, and construct business time window and electronic fence by analyzing the driving habits of vehicles. Furthermore, we propose a probabilistic logic based data segmentation method (PLDSM) which not only helps finding all the business points but also assists in inferring the business affair categories. An efficient
In this paper, we propose a dynamically low-dimensional approximation method to solve a class of time-dependent multiscale stochastic diffusion equations. In Cheng etal. (2013) a dynamically bi-orthogonal (DyBO) method was developed to explore low-dimensional structures of stochastic partial differential equations (SPDEs) and solve them efficiently. However, when the SPDEs have multiscale features in physical space, the original DyBO method becomes expensive. To address this issue, we construct multiscale basis functions within the framework of generalized multiscale finite element method (GMsFEM) for dimension reduction in the physical space. To further improve the accuracy, we also perform online procedure to construct online adaptive basis functions. In the stochastic space, we use the generalized polynomial chaos (gPC) basis functions to represent the stochastic part of the solutions. Numerical
In this paper, we propose a model reduction method for solving multiscale elliptic PDEs with random coefficients in the multiquery setting using an optimization approach. The optimization approach enables us to construct a set of localized multiscale data-driven stochastic basis functions that give an optimal approximation property of the solution operator. Our method consists of the offline and online stages. In the offline stage, we construct the localized multiscale data-driven stochastic basis functions by solving an optimization problem. In the online stage, using our basis functions, we can efficiently solve multiscale elliptic PDEs with random coefficients with relatively small computational costs. Therefore, our method is very efficient in solving target problems with many different force functions. The convergence analysis of the proposed method is also presented and has been verified by the numerical simulations.
We develop a multiscale tailored finite point method (MsTFPM) for second order elliptic equations with rough or highly oscillatory coefficients. The finite point method has been tailored to some particular properties of the problem, so that it can capture the multiscale solutions using coarse meshes without resolving the fine scale structure of the solution. Several numerical examples in one-and two-dimensions are provided to show the accuracy and convergence of the proposed method. In addition, some analysis results based on the maximum principle for the one-dimensional problem are proved.
In this paper, we present an adaptive, analysis of variance (ANOVA)-based data-driven stochastic method (ANOVA-DSM) to study the stochastic partial differential equations (SPDEs) in the multi-query setting. Our new method integrates the advantages of both the adaptive ANOVA decomposition technique and the data-driven stochastic method. To handle high-dimensional stochastic problems, we investigate the use of adaptive ANOVA decomposition in the stochastic space as an effective dimension-reduction technique. To improve the slow convergence of the generalized polynomial chaos (gPC) method or stochastic collocation (SC) method, we adopt the data-driven stochastic method (DSM) for speed up. An essential ingredient of the DSM is to construct a set of stochastic basis under which the stochastic solutions enjoy a compact representation for a broad range of forcing functions and/or boundary conditions
The numerical solution of the one-dimensional KleinGordon equation on an unbounded domain is analyzed in this paper. Two artificial boundary conditions are obtained to reduce the original problem to an initial boundary value problem on a bounded computational domain, which is discretized by an explicit difference scheme. The stability and convergence of the scheme are analyzed by the energy method. A fast algorithm is obtained to reduce the computational cost and a discrete artificial boundary condition (DABC) is derived by the <i>Z</i>-transform approach. Finally, we illustrate the efficiency of the proposed method by several numerical examples.
The numerical solution of the one-dimensional nonlinear Klein-Gordon equation on an unbounded domain is studied in this paper. Split local absorbing boundary (SLAB) conditions are obtained by the operator splitting method, then the original problem is reduced to an initial boundary value problem on a bounded computational domain, which can be solved by the finite difference method. Several numerical examples are provided to show the advantages and effectiveness of the given method, and some interesting collision behaviors are also observed.
This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper , we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O ((m/N p) 3) for a
We propose a dynamically bi-orthogonal method (DyBO) to solve time dependent stochastic partial differential equations (SPDEs). The objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi-orthogonal basis. It is well-known that the KarhunenLoeve expansion (KLE) minimizes the total mean squared error and gives the sparsest representation of stochastic solutions. However, the computation of the KL expansion could be quite expensive since we need to form a covariance matrix and solve a large-scale eigenvalue problem. The main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the KL expansion. Unlike other reduced model methods, our method constructs the reduced basis on-the-fly without the need to form the
The curvature estimates for k curvature equations with general right-hand sides is a longstanding problem. In this paper, we completely solve the problem when k . We also discuss some applications of our estimates.
We give a new proof of a classical uniqueness theorem of Alexandrov  using the weak uniqueness continuation theorem of BersNirenberg . We prove a version of this theorem with the minimal regularity assumption: the spherical Hessians of the corresponding convex bodies as Radon measures are nonsingular.
We establish an interior C 2 estimate for k+ 1 convex solutions to Dirichlet problems of k-Hessian equations. We also use such estimate to obtain a rigidity theorem for k+ 1 convex entire solutions of k-Hessian equations in Euclidean space.