Uniform entropy bound for Ricci shrinkers with bounded curvature.
problem Bounding entropy for Ricci shrinkers with specific curvature constraints.
method Establishing uniform entropy bounds for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
result Uniform entropy bound for simply connected Ricci shrinkers with a finite second homotopy group and uniform curvature bounds.
Paper finds maximum curvature of Bézier-spline curves.
problem Finding maximum curvature of Bézier-spline curves.
method Modified B-spline solutions for inverse interpolation problem.
result Determined maximum curvature of Bézier-spline curves.
The paper proves uniformization for specific curvature types on manifolds.
problem Uniformizing fourth order conformal curvature on Riemannian manifolds.
method Proving existence of conformal deformations for specific curvature conditions.
result Existence of conformal deformations for positive Yamabe invariant and total Q-curvature.
The paper proves conditions for existence of constant scalar curvature Kähler metrics with cone singularities.
problem Existence of constant scalar curvature Kähler metrics with cone singularities.
method Log K-polystability and G-uniform log K-stability are established. result Uniform log K-stability is achieved for normal varieties. We prove a uniform extension result for contracting maps defined on subsets of Hadamard manifolds subject to curvature bounds.
Uniform bounds on ends for non-branching CD spaces with nonnegative curvature outside a compact set.
problem Bounding the number of ends of non-branching CD spaces with nonnegative curvature outside a compact set.
method Adapting Z.-D. Liu's work to prove a ball covering property.
result Uniform bounds on the number of ends of such spaces.
Uniform curvature bounds for regularized metrics with bounds on Ricci tensor and injectivity radius.
problem Bounding curvature of regularized metrics with constraints on Ricci tensor and injectivity radius.
method Mollification of riemannian metrics, uniform W2,p-harmonic radius bounds, Ricci tensor bounds, injectivity radius bounds. result Uniform estimate on the change of sectional curvature for regularized metrics.
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties of the smooth limit surface and the shape regularity of the discrete net.
Abstract machinery finds obstructions to uniform positive scalar curvature.
problem Finding obstructions to uniform positive scalar curvature.
method Coarse index theory and embedding submanifolds.
result Abstract machinery constructs wrong way maps on K-theory. On Riemannian manifolds of dimension 4, for prescribed scalar curvature equation, under lipschitzian condition on the prescribed curvature, we have an uniform estimate for the solutions of the equation if we control their minimas.
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Kähler-Ricci flow on Kähler manifolds converges to negative Kodaira dimension
problem Convergence of scalar curvature in Kähler-Ricci flow
method Uniform μ-entropy or uniform Sobolev inequality result Scalar curvature converges to negative Kodaira dimension
New proof of uniformization for hyperbolic foliations.
problem Uniformization of foliated spaces by surfaces of hyperbolic type.
method Laminated Ricci flow to find a conformally equivalent metric with constant curvature -1.
result Existence of a laminated Riemannian metric with leaves of constant Gaussian curvature -1.
The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.
problem Finding decorated piecewise hyperbolic metrics with prescribed combinatorial curvature.
method Introduced combinatorial α-Ricci flow with surgery to handle potential singularities and prove longtime existence and convergence.
result Existence of decorated piecewise hyperbolic metrics with prescribed combinatorial α-curvature.
We prove that complete Riemannian manifolds with polynomial growth and Ricci curvature bounded from below, admit uniform Poincaré inequalities. A global, uniform Poincaré inequality for horospheres in the universal cover of a closed, n-dimensional Riemannian manifold with pinched negative sectional curvature follows …
Study shows uniform bounds on torsion and curvature for Chern-Ricci flow solutions.
problem Understanding singularity types in long-time solutions of Chern-Ricci flow.
method Extended results from Kähler-Ricci flow to Chern-Ricci flow, focusing on uniform bounds on torsion and curvature.
result Uniform bounds on torsion and curvature for solutions starting from metrics of the same ∂∂ˉ class. Uniform convergence of metrics on surfaces with bounded curvature measures proved.
problem Proving uniform convergence of metrics on Alexandrov surfaces with bounded integral curvature.
method Weak convergence of measures and analytic approximation of metrics.
result Uniform convergence of metrics on Alexandrov surfaces proved.
Uniform eigenvalue bounds for Hodge Laplacian on manifolds with Ricci curvature and injectivity radius bounds.
problem Establishing uniform bounds for eigenvalues of the Hodge Laplacian on manifolds with specific geometric constraints.
method Using uniform bounds on Ricci curvature, injectivity radius, and diameter, we derive eigenvalue estimates for the Hodge Laplacian.
result Uniform upper bounds for eigenvalues of the Hodge Laplacian on differential forms on manifolds with given geometric constraints.
Uniform bounds for eigenvalues of Hodge Laplacian on manifolds with lower Ricci curvature.
problem Establishing bounds for eigenvalues of Hodge Laplacian under lower Ricci curvature.
method Using geometric assumptions including lower Ricci curvature, injectivity radius, and diameter bounds.
result Uniform eigenvalue bounds for the Hodge Laplacian and connection Laplacian.
Mean curvature flow with uniform bounds on curvature and its gradient
problem Mean curvature flow
method Uniform bounds on curvature and its gradient
result Smooth extension past singular time
Study on curvature invariants near singularities of wavefronts.
problem Conditions for extendibility and boundedness of curvature invariants.
method Investigation of Gaussian curvature, Mean curvature, and principal curvatures near singularities.
result Relationship between convergence to infinity and uniform approximation of fronts.
In this article we show that for every finite area hyperbolic surface X of type (g,n) and any harmonic Beltrami differential μ on X, then the magnitude of μ at any point of small injectivity radius is uniform bounded from above by the ratio of the Weil-Petersson norm of μ over the square root of the systole…
Proves constant scalar curvature Kähler metrics are very general.
problem Existence of constant scalar curvature Kähler metrics on smooth polarized varieties.
method Combining uniform arc K-stability and algebraic properties in families.
result The constant scalar curvature Kähler locus is very general.
The paper introduces a new discretization of Gaussian curvature on surfaces.
problem Discretizing Gaussian curvature on surfaces with conic singularities.
method Discrete conformal theory and variational principles with constraints.
result Established a discrete uniformization theorem for surfaces with non-positive Euler number.
We give some uniform estimates for constant mean curvature solutions of the conformal vacuum Einstein constraint equations on compact manifolds. Existence of those solutions was given in a paper by J. Isenberg.
Study shows limits of metrics with positive scalar curvature on spheres.
problem Non-negativity of scalar curvature is not preserved under certain limits.
method Examined metrics conformal to the round metric on Sn for n≥4. result Any conformal metric to the round metric on Sn for n≥4 can be a limit of metrics with positive scalar curvature. Smooths metrics on manifolds with curvature bounds and injectivity radius constraints.
problem Smooth metrics on manifolds with curvature and injectivity constraints.
method Bi-Lipschitz smoothing with controlled smoothing and volume lower bounds.
result Proves existence of smooth metrics with curvature bounds and injectivity radius constraints.
Starting with a model conical Kähler metric, we prove a uniform scalar curvature bound for solutions to the conical Kähler-Ricci flow assuming a semi-ampleness type condition on the twisted canonical bundle. In the proof, we also establish uniform estimates for the potentials and their time derivatives.
A geometric framework for metrics of maximal acceleration which is applicable to large proper accelerations is discussed, including a theory of connections associated with the geometry of maximal acceleration. In such a framework it is shown that the uniform bound on the proper maximal acceleration implies an uniform b…
In this paper we study a boundary value problem for the Ricci flow in the two dimensional ball endowed with a rotationally symmetric metric. We show short and long time existence results. We construct families of metrics for which the flow uniformizes the curvature along a sequence of times. Finally we show that if the…
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu−e2u=K0(z) on general open surfaces. A few oth…
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
In this paper, we introduce a parameterized discrete curvature (α-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uni…
Study eigenvalues for special curvature equations on star-shaped surfaces.
problem Eigenvalue problem for prescribed curvature equations on star-shaped, k-convex hypersurfaces.
method Established existence of a unique eigenvalue and hypersurface through uniform estimates in p for Lp-type equations.
result Existence of a unique eigenvalue and associated hypersurface under certain conditions.
Study compact sequences of warped product circles over spheres with nonnegative scalar curvature.
problem Compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
method Proved subsequence convergence to a W1,p Riemannian metric for all p<2 with nonnegative scalar curvature in the distributional sense. result Proved compactness of sequences of warped product circles over spheres with nonnegative scalar curvature.
This paper examines limits of Riemannian 2-manifolds with bounded curvature.
problem Understanding the limits of Riemannian 2-manifolds with bounded curvature.
method Uniform semi-locally 1-connected sequences of closed connected Riemannian 2-manifolds with bounded total absolute curvature.
result Description of Gromov-Hausdorff limits of the sequences.
We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower…
In this paper we describe the topology of 4-dimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature and with a uniform upper bound of diameter which collapse to metric spaces of lower dimensions. This enables us to understand the set of homeomorphism classes of closed orient…
For sequences of warped product metrics on a 3-torus satisfying the scalar curvature bound Rj≥−j1, uniform upper volume and diameter bounds, and a uniform lower area bound on the smallest minimal surface, we find a subsequence which converges in both the Gromov-Hausdorff and the Sormani-Wenger Intrin…
New approach to nematic fields on surfaces, relaxing uniformity to quasi-uniformity.
problem Identifying least distorted nematic fields on generic surfaces.
method Relaxing the notion of uniformity into quasi-uniformity and proving parallel transport by geodesics.
result All quasi-uniform fields are parallel transported by the geodesics of the surface.
We study the equation Δgu−4(n−1)n−2R(g)u+Kup=0(1+ζ≤p≤n−2n+2) on locally conformally flat compact manifolds (Mn,g). We prove the following: (i) When the scalar curvature R(g)>0 and the dimension n≥4, under suitable conditions on K, all positive solutions u have uniform u…
We introduce a norm on the space of test configurations, which we call the minimum norm. We conjecture that uniform K-stability with respect to this norm is equivalent to the existence of a constant scalar curvature Kähler metric. This notion of uniform K-stability is analogous to coercivity of the Mabuchi functional. …
The paper provides estimates for higher-order Ricci curvature along Kähler-Ricci flows.
problem Estimating higher-order curvature along Kähler-Ricci flows on compact Kähler manifolds.
method Proving uniform bounds for Ricci curvature and scalar curvature in various orders and norms.
result A geometric obstruction causes a specific third-order derivative of Ricci curvature to blow up at rate et/2. We show that complete uniform visibility manifolds of finite volume with sectional curvature −1≤K≤0 have positive simplicial volumes. This implies that their minimal volumes are non-zero.
The paper simplifies K-stability conditions for spherical varieties.
problem K-stability of polarized spherical varieties.
method Expressed K-stability in combinatorial terms, provided sufficient conditions.
result G-uniform K-stability provides a checkable condition for existence of constant scalar curvature metrics.
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.
We establish a uniform estimate for the injectivity radius of the past null cone of a point in a general Lorentzian manifold foliated by spacelike hypersurfaces and satisfying an upper curvature bound. Precisely, our main assumptions are, on one hand, upper bounds on the null curvature of the spacetime and the lapse fu…
The study extends curvature bounds to non-smooth spaces and proves stability of mean curvature.
problem Proving curvature bounds in non-smooth spaces.
method Extending results from smooth Riemannian manifolds to non-smooth RCD spaces.
result Stability of mean curvature bounds under uniform convergence.