Study translating solitons in Minkowski space with prescribed Gauss image.
problem Second boundary value problem for mean curvature flow in Minkowski space.
method Construct translating solitons with prescribed Gauss image.
result Constructing translating solitons with prescribed Gauss image in Minkowski space.
The paper finds convex hypersurfaces with specific curvature properties.
problem Finding convex hypersurfaces with prescribed Hessian curvatures and Gauss images.
method Used novel C2 boundary estimates based on orthogonal invariance and infinitesimal rotations. result Proved existence of strictly convex graphic hypersurfaces with prescribed k-Hessian curvatures. New real algebraic maps with prescribed images and compositions are constructed locally like moment maps.
problem Constructing real algebraic maps with specific properties and compositions.
method Explicit construction of real algebraic hypersurfaces and maps with prescribed images and compositions.
result Explicit families of functions represented as compositions of constructed maps with canonical projections.
Paper finds unique solutions for curved surfaces with specific gradient.
problem Existence of curved surfaces with specific gradient.
method Second boundary value problem of constant mean curvature equations.
result Unique convex solutions for constant mean curvature equations.
We study that the n−graphs defining by smooth map $f:\Om\subset \ir{n}\to \ir{m}, m\ge 2,$ in $\ir{m+n}$ of the prescribed mean curvature and the Gauss image. We derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le\f{C}{R^2}$$ under the dimension limitations and the Gauss image restrictions. If there is no…
We consider a classical N. Steenrod's problem on realization of homology classes by images of the fundamental classes of manifolds. It is well-known that each integral homology class can be realized with some multiplicity as an image of the fundamental class of a manifold. Our main result is an explicit purely combinat…
Geometric approach to meromorphic differentials' periods and their holonomy representations.
problem Characterizing representations of meromorphic differentials' periods.
method Constructing translation structures with prescribed holonomy.
result Generalization of Haupt's classical result to meromorphic differentials.
Study Palais-Smale sequences for Ricci curvature on homogeneous spaces.
problem Characterize Palais-Smale sequences for prescribed Ricci curvature.
method Complete description of divergent sequences on compact homogeneous spaces.
result Existence of saddle points on generalized Wallach and flag manifolds.
The paper proves the existence of specific spacelike hypersurfaces in Minkowski space.
problem Proving the existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. method Analyzing hypersurfaces in Minkowski space, proving existence through curvature and Gauss map properties.
result Existence of smooth, entire, strictly convex, spacelike hypersurfaces with constant σk curvature. We use a variational principle to prove an existence and uniqueness theorem for planar weighted Delaunay triangulations (with non-intersecting site-circles) with prescribed combinatorial type and circle intersection angles. Such weighted Delaunay triangulations may be interpreted as images of hyperbolic polyhedra with …
Constructs polyhedral chains with prescribed tangent plane distributions.
problem Constructing polyhedral chains with specific tangent plane distributions.
method Explicit construction of polyhedral chains that approximate prescribed measures on Grassmannian.
result Polyconvexity is equivalent to quasiconvexity of associated Q-integrands under certain conditions.
Study on curvature image iterations converging to solutions of Minkowski problems.
problem Solving Lp Minkowski problems with prescribed data. method Iterations of curvature image operators Λpφ applied to convex bodies. result Iterations of curvature image operators converge to fixed points under certain conditions.
We propose and analyze a constrained level-set method for semi-automatic image segmentation. Our level-set model with constraints on the level-set function enables us to specify which parts of the image lie inside respectively outside the segmented objects. Such a-priori information can be expressed in terms of upper a…
The paper proves the existence and uniqueness of certain spacelike hypersurfaces with specific curvature and boundary conditions.
problem Existence and uniqueness of convex, entire, spacelike hypersurfaces with constant σk curvature. method Investigation of hypersurfaces with prescribed set of lightlike directions and perturbation on the ideal boundary at infinity.
result Existence and uniqueness of complete entire spacelike constant σk curvature hypersurfaces with prescribed lightlike directions and perturbation. Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…
We study projective structures on a surface having poles of prescribed orders. We obtain a monodromy map from a complex manifold parameterising such structures to the stack of framed PGL2(C) local systems on the associated marked bordered surface. We prove that the image of this map is contained in…
Spherical quadrilaterals classified based on geometric properties.
problem Classifying spherical quadrilaterals up to isometry.
method Developing map and condition of genericity.
result Space of quadrilaterals with prescribed angles consists of finitely many open curves.
fSGLD optimizes deep learning by favoring flat regions in the loss landscape.
problem Understanding and improving the behavior and generalization of deep learning algorithms.
method Flatness-Aware Stochastic Gradient Langevin Dynamics (fSGLD) that biases learning towards flat basins.
result fSGLD targets a flatness-biased Gibbs distribution with explicit excess risk guarantees.
Study on the convergence rate of prescribed scalar curvature flow.
problem Prescribing scalar curvature on manifolds.
method Inspired by Yamabe flow convergence rate study, analyze the prescribed scalar curvature flow convergence rate.
result Determine the convergence rate of the prescribed scalar curvature flow.
Develops a new method to create object models from medical images.
problem Variability in anatomical structures and textures limits observer performance.
method Progressive Growing AmbientGAN (ProAmGAN) for creating stochastic object models from medical imaging measurements.
result Demonstrates the effectiveness of ProAmGAN in creating realistic object models.
Study symmetry groups and curves from sums of exponentials.
problem Understanding the geometry and symmetry of curves from sums of exponentials.
method Analysis of symmetry groups, winding numbers, and parametrization of the unit circle.
result Unified method for constructing curves with specific properties.
This paper describes connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. Unlike the case for unmarked Riemann surfaces, we find there can be many connected components, distinguished by roots of the cotangent bundle of the surface. In the c…
Proves optimal regularity for sphere minimizers in 3-sphere.
problem Finding optimal regularity for sphere minimizers.
method Proves C1,1 regularity for minimizers of prescribed mean curvature over isotopy classes. result Proves optimal C1,1 regularity for minimizers. Proves existence and uniqueness of Killing graphs with prescribed curvature.
problem Existence and uniqueness of Killing graphs with prescribed curvature.
method Proves existence and uniqueness of Killing graphs with prescribed mean curvature considering non-constant functions.
result Existence and uniqueness of Killing graphs with prescribed curvature.
Develops variational approach for Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
problem Existence and characterization of Kähler-Einstein metrics with prescribed singularities on Fano manifolds.
method Variational approach, algebraic approximation of singularities, function α_ω, continuity method.
result Many K-stable manifolds admit all possible Kähler-Einstein metrics with prescribed singularities.
Paper estimates curvature of convex hypersurfaces with prescribed curvature.
problem Estimating curvature of p-convex hypersurfaces with prescribed curvature. method Establishes curvature estimates for p-convex hypersurfaces in Rn+1 with p≥2n. result Proves existence of star-shaped hypersurface of prescribed curvature and interior C2 estimates. Theory proves existence of hypersurfaces with prescribed curvature.
problem Existence of hypersurfaces with prescribed mean curvature in noncompact manifolds.
method Developed min-max theory for noncompact manifolds.
result Proved existence of closed and finite area hypersurfaces.
Introduce a flow for prescribed Hermitian-Yang-Mills tensors
problem Solve the prescribed Hermitian-Yang-Mills tensor equation
method Long-time convergence of the flow
result Uniform C0-estimate of the flow We develop a compactness result near the boundary for families of locally convex immersions. We also develop a mod 2 degree theory for immersion of constant (and prescribed) Gaussian curvature with prescribed boundary. These are then used to solve the Plateau problem for immersions of constant (and prescribed) Gaussian…
Pixel-space diffusion models outperform latent models on high-resolution image synthesis.
problem Efficiency and quality trade-off in high-resolution image synthesis.
method Sigmoid loss-weighting, simplified architecture, and resolution scaling.
result Achieved 1.5 FID on ImageNet512, new SOTA results on other datasets.
This work detects anomalous clients in federated learning to prevent their adverse impacts.
problem Detecting and preventing anomalous client behaviors in federated learning systems.
method Generates low-dimensional surrogates of model weight vectors and uses them for anomaly detection.
result The proposed detection-based approach significantly outperforms conventional defense-based methods.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and vo…
Study convex capillary hypersurfaces with prescribed curvature in a spherical cap.
problem Prescribed curvature problem for convex capillary hypersurfaces.
method Reformulated as Hessian quotient equation with Robin boundary condition.
result Existence of strictly convex capillary hypersurface with prescribed curvature.
Paper solves degenerated circle pattern metric problem in spherical geometry.
problem Existence and rigidity of (degenerated) circle pattern metrics with prescribed total geodesic curvatures.
method Defined prescribed combinatorial Ricci flows and studied their convergence.
result First degenerated result for total geodesic curvatures in spherical background geometry.
The paper constructs real algebraic functions with specific singularities and preimages.
problem Constructing real algebraic functions with prescribed singular points and preimages.
method Generalizing moment-like maps and Morse-Bott functions to real algebraic settings.
result Explicit construction of real algebraic functions with exactly one singular value and prescribed preimages.
Paper solves Dirichlet problem for p-convex hypersurfaces with curvature constraints.
problem Solving the Dirichlet problem for p-convex hypersurfaces with prescribed curvature. method Proved existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition, obtained an interior curvature estimate.
result Existence of a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition.
Study on prescribing Ricci curvature on compact Lie groups.
problem Prescribing Ricci curvature in naturally reductive metrics on compact Lie groups.
method Derive necessary and sufficient conditions for solvability.
result Provide a series of examples.
Study on t-graphs with prescribed mean curvature in Heisenberg groups.
problem Existence and uniqueness of t-graphs with prescribed mean curvature. method Characterization of classical solutions without Dirichlet boundary data, conditions for uniqueness, approximation technique for non-constant mean curvature.
result Conditions for existence and uniqueness of t-graphs in Heisenberg groups. We prove that on any compact complex manifold one can find Gauduchon metrics with prescribed volume form. This is equivalent to prescribing the Chern-Ricci curvature of the metrics, and thus solves a conjecture of Gauduchon from 1984.
The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.
The paper constructs surfaces with prescribed mean curvature in a specific space.
problem Finding surfaces with a given mean curvature in a particular geometric space.
method Phase plane analysis to construct entire rotational graphs and catenoid-type surfaces.
result Classification result for surfaces with linearly prescribed mean curvature.
The paper solves a curvature problem on a ball's surface near constant values.
problem Prescribing almost constant curvatures on a manifold with boundary.
method Perturbative approach and ansatz by Han and Li.
result New existence results for conformal metrics when curvatures are near constants.
Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.
problem Short time existence and long time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
method Finding sufficient conditions for short time existence and discussing long time existence and convergence.
result Sufficient conditions for short time existence of prescribed mean curvature flow on noncompact spacelike hypersurfaces.
In his paper "Shapes of Polyhedra and Triangulations of the Sphere", Thurston found that the set of shapes of convex polyhedra with prescribed cone-deficits has a complex hyperbolic structure. Inspired by his work, this paper studies the set of shapes of centrally symmetric octahedra with prescribed cone-deficits. We s…
In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data an…
Proves existence of graph on torus with prescribed curvature.
problem Existence of graphs with prescribed mean curvature on torus.
method Proves existence using graph theory and curvature conditions.
result Existence of a graph on the n-dimensional torus with prescribed curvature.
Optical flow refers to the visual motion observed between two consecutive images. Since the degree of freedom is typically much larger than the constraints imposed by the image observations, the straightforward formulation of optical flow as an inverse problem is ill-posed. Standard approaches to determine optical flow…
In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.