Study proves global existence of weak solutions for mean curvature flow with transport and forcing terms.
problem Existence of weak solutions for mean curvature flow with non-smooth terms.
method Used modified Allen-Cahn equation with properties like monotonicity formula to prove existence.
result Global existence of weak solutions for the mean curvature flow with transport and forcing terms.
New formulas with quadratic curvature terms on Kähler manifolds for Hodge number estimates.
problem Estimating Hodge numbers under weak curvature conditions.
method Established new Bochner-Kodaira formulas with quadratic curvature terms.
result Derivation of Weitzenböck-Bochner-Kodaira formulas with quadratic curvature terms on compact Kähler manifolds.
Proves global existence of maps with curvature term on expanding spacetimes.
problem Global existence of Dirac-wave maps with curvature term on expanding spacetimes.
method Proves global existence with small initial data on globally hyperbolic manifolds with growth condition.
result Global existence of maps proved for small initial data.
Flow of planar curves with curvature and capacity potential.
problem Geometric flow of planar curves with curvature and capacity potential.
method Curvature flow with a nonlocal term (normal derivative of capacity potential).
result Long term existence and large time asymptotics established under convexity condition.
Integral of scalar curvature equals a volume ratio term on certain 3D manifolds.
problem Integral of scalar curvature on manifolds with a pole.
method Asymptotic scaling invariant integral of scalar curvature equals a term determined by asymptotic volume ratio.
result Integral of scalar curvature equals a volume ratio term.
Formula calculates mass using cube faces and edges.
problem Measuring mass of 3-manifolds.
method Cube faces and edges, mean curvature, dihedral angle, geodesic curvature, angle defect.
result Mass formula connects to Gromov's theory and Gauss-Bonnet theorem.
We study several geometric and analytic aspects of Dirac-harmonic maps with curvature term from closed Riemannian surfaces.
New flow method solves Christoffel-Minkowski problem.
problem Solving Christoffel-Minkowski problem.
method Entropy preserving curvature flow with global term.
result Entropy preserving flow solves Christoffel-Minkowski problem.
In this paper, we consider the mean curvature flow of convex hypersurfaces in Euclidean spaces with a general forcing term. We show that the flow may shrink to a point in finite time if the forcing term is small, or exist for all times and expand to infinity if the forcing term is large enough. The flow can also conver…
New proof of isoperimetric inequality under Ricci curvature bounds.
problem Proving isoperimetric inequalities in spaces with lower Ricci curvature bounds.
method Using Minkowski content and perimeter in metric measure spaces with curvature dimension condition.
result Isoperimetric inequality and Cheeger constant results hold in spaces with lower Ricci curvature bounds.
The paper proves Buser's inequality for graphs with positive curvature.
problem Establishing bounds for eigenvalues in terms of Cheeger constants for infinite graphs.
method Proved Buser's inequality for graphs with Ricci curvature lower bounds and derived a lower bound on Cheeger constant in terms of positive curvature.
result Graphs with positive curvature are finite, especially for unbounded Laplacians.
We show that any universal curvature identity which holds in the Riemannian setting extends naturally to the pseudo-Riemannian setting. Thus the Euh-Park-Sekigawa identity also holds for pseudo-Riemannian manifolds. We study the Euler-Lagrange equations associated to the Chern-Gauss-Bonnet formula and show that as in t…
Upper bound for total mean curvature of spin fill-ins is proven.
problem Bounding the total mean curvature of spin Riemannian manifolds.
method Proving Gromov's conjecture for spin manifolds with specific conditions.
result Explicit upper bounds for total mean curvature are derived under various conditions.
HR in 8D encodes unique conformal gravity with negative curvature.
problem Holographic Renormalisation in 8D Einstein Gravity.
method Relating HR to Topological Regularisation and adding the Euler term.
result The unique conformal gravity theory reproduces the polynomial and cancels divergent terms.
New curvature condition ensures projectivity of Kähler manifolds.
problem Characterize projectivity of Kähler manifolds using curvature.
method Introduced a new curvature condition and proved it implies projectivity.
result Positive 2nd scalar curvature guarantees projectivity of Kähler manifolds.
Paper finds flag curvature of submanifolds in Randers-Minkowski space using Zermelo data.
problem Characterizing submanifolds with scalar flag curvature in Randers-Minkowski spaces.
method Expresses flag curvature in terms of Zermelo data invariants.
result Proves any h-flat hypersurface has scalar F-flag curvature and conformally flat metric.
New algebraic characterization of sectional curvature bounds using Weitzenböck formulae.
problem Establishing sectional curvature bounds using curvature terms in Weitzenböck formulae.
method Introducing a symmetric analogue of the Kulkarni-Nomizu product and applying the Bochner technique.
result New insights into the Hopf Conjecture for closed 4-manifolds with indefinite intersection form and positive sectional curvature.
Proves uniqueness of blowups for forced mean curvature flow.
problem Proving uniqueness of blowups for forced mean curvature flow.
method Adapting methods from Euclidean space mean curvature flow to handle forcing term and blow-up limits.
result Uniqueness of tangent cones for forced mean curvature flow at self-shrinkers and cylindrical self-shrinkers.
Estimates submanifold diameters in curved spaces.
problem Estimating the intrinsic diameter of submanifolds in curved spaces.
method Using mean curvature field integrals and boundary lengths.
result Diameter estimates for submanifolds in curved spaces.
We prove qualitative estimates on the total curvature of closed minimal hypersurfaces in closed Riemannian manifolds in terms of their index and area, restricting to the case where the hypersurface has dimension less than seven. In particular, we prove that if we are given a sequence of closed minimal hypersurfaces of …
The paper connects curvature data to polynomial coefficients in gluing formulas.
problem Understanding polynomial coefficients in gluing formulas for zeta-determinants.
method Expressing coefficients of a polynomial in terms of scalar and principal curvatures of a 2D hypersurface.
result Coefficients of the polynomial are expressed in terms of curvature data.
Approximates nonlocal curvature of curves using splines.
problem Approximating nonlocal curvature of planar curves.
method Extending nonlocal curvature definition, using incomplete beta function, and linear interpolating spline approximation.
result Nonlocal curvature of a planar curve can be approximated by a spline.
We prove universal recursive formulas for Branson's Q-curvatures in terms of respective lower-order Q-curvatures, lower-order GJMS-operators and holographic coefficients.
Expresses curvature and connection of one metric in terms of another.
problem Expressing curvature and connection of a metric in terms of another.
method Generalizes coordinate expressions for Riemannian manifolds.
result Formulas for curvature and connection of a metric in terms of another metric.
We relate the positivity of the curvature term in the Weitzenbock formula for the Laplacian on p-forms on a complete manifold to the existence of bounded and L2 harmonic forms. In the case where the manifold is the universal cover of a compact manifold, we obtain topological and geometric information about the compa…
The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.
problem Proving Li-Yau inequalities and modified logarithmic Sobolev inequalities for reversible Markov chains.
method Introducing the CDΥ(κ,F) condition and deriving entropy-information inequalities. result Derives functional inequalities relating entropy to Fisher information.
Sharp bounds on mean curvature and geodesic lengths in convex hypersurfaces.
problem Finding sharp bounds on total mean curvature of convex hypersurfaces.
method Sharp lower bounds for mean width and Birkhoff invariant, characterizing spheres.
result Generalization of Álvarez Paiva's result to convex hypersurfaces.
A general formulation of zero curvature connections in a principle bundle is presented and some applications are discussed. It is proved that a related connection based on a prolongation in an associated bundle remains zero curvature as well. It is also shown that the connection coefficients can be defined so that the …
Study inverse mean curvature flow on non-compact hypersurfaces, proving long-term existence and characterizing maximal time.
problem Evolution of non-compact convex hypersurfaces in Rn+1 by inverse mean curvature. method Establish long-term existence via pointwise mean curvature estimate and viscosity solutions for strict convexity.
result Characterization of maximal time of existence in terms of tangent cone at infinity.
We show that under suitable non-degeneracy conditions, complete gradient flow lines of the scalar curvature functional of a riemannian manifold perturb into eternal forced mean curvature flows with large forcing term.
Study proves long-term flow for special submanifolds.
problem Long-term behavior of spacelike submanifolds.
method Proves long-time existence for mean curvature flow of spacelike submanifolds under specific curvature conditions.
result Proves long-time existence for mean curvature flow of spacelike submanifolds.
The paper extends curvature concepts to surfaces in normed spaces.
problem Extending curvature concepts to surfaces in normed spaces.
method Using Birkhoff orthogonality and surface areas, the paper characterizes Minkowski Gaussian curvature.
result Characterizations and generalizations of classical theorems for curvature in Minkowski spaces.
Directly proves Brioschi formula for Gaussian curvature.
problem Express Gaussian curvature in terms of local coordinates.
method Elementary proof without Christoffel symbols.
result Directly derived Brioschi formula for Gaussian curvature.
Study on modified Ricci curvature on graphs, proving rigidity and deriving formulas.
problem Understanding Ricci curvature on graphs, especially for specific graph types.
method Introduced modified Ricci curvature, established rigidity theorem, derived formulas for strongly regular graphs.
result Rigidity theorem for complete graphs and explicit formulas for strongly regular graphs.
We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive p-curvature. The p-curvature was defined and studied by the second author. It turns out that positivity of p-curvature could be preserved under surgeries of codimension at least p+3. This gives a key to …
We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of curvature is a second order system of partial differential equations which we sho…
We study different notions of Riemannian curvatures: The p-curvatures which interpolate between the scalar curvature and the sectional curvature, the Gauss-Bonnet-Weyl curvatures form another interpolation from the scalar curvature to the Gauss-Bonnet integrand. We bring out the (p,q)-curvatures, which incorporate …
Explicit computation of Kontsevich weights for symplectic Poisson structures.
problem Computing weights of Kontsevich graphs in symplectic Poisson structures.
method Detailed explicit computation using hypergeometric functions and simpler formulas.
result Explicit expressions for curvature weights and their simplification in cotangent bundles.
Quantitative estimate for curvature in mean curvature flow.
problem Estimating curvature in mean curvature flow.
method Proving a curvature estimate for smooth convex ancient flows.
result Curvature grows at most quadratically in terms of rescaled extrinsic distance.
This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.
problem Analyzing the Bochner formula for Riemannian flows and deriving eigenvalue estimates.
method The approach involves studying the curvature term in the Bochner-Weitzenb{ö}ck formula of the basic Laplacian on M, splitting it into two parts, and establishing eigenvalue estimates.
result Established an eigenvalue estimate of the basic Laplacian on basic forms, and discussed the limiting case of the estimate.
We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…
Lower bounds on curvature integral for manifolds with curvature constraints.
problem Bounding curvature integrals under curvature constraints.
method Proving a lower bound for the curvature integral using dimension, upper curvature bounds, and injectivity radius.
result Uniformly bounded below integral of scalar curvature.
We consider an asymptotically flat Riemannian spin manifold of positive scalar curvature. An inequality is derived which bounds the Riemann tensor in terms of the total mass and quantifies in which sense curvature must become small when the total mass tends to zero.
Study on bending energy of surfaces with curvature concentration, deriving new lower bounds.
problem Analyzing the Willmore energy of surfaces with curvature concentration.
method Using isoperimetric inequalities and framed loops, derive new lower bounds for the bending energy.
result Optimal blowup rates of the Willmore energy when curvature is concentrated.
We establish a min-max estimate on the volume width of a closed Riemannian manifold with nonnegative Ricci curvature. More precisely, we show that every closed Riemannian manifold with nonnegative Ricci curvature admits a PL Morse function whose level set volume is bounded in terms of the volume of the manifold. As a c…
The study characterizes Finsler metrics and proves their rigidity.
problem Characterizing and proving rigidity of Busemann convex Finsler metrics.
method Proving Finsler metrics are nonpositively curved if and only if they are affinely equivalent to a Riemannian metric of nonpositive sectional curvature.
result Finsler metrics are precisely Berwald metrics of nonpositive flag curvature.
Flow preserves quermassintegrals, converging to a geodesic sphere.
problem Volume preservation issue in sphere mean curvature flow.
method Introduced a mean curvature flow with a global term to keep quermassintegrals fixed.
result Flow exists for all times and converges to a geodesic sphere.
We study harmonic maps from Riemannian manifolds into arbitrary non-positively curved and CAT(-1) metric spaces. First we discuss the domain variation formula with special emphasis on the error terms. Expanding higher order terms of this and other formulas in terms of curvature, we prove an analogue of the Eels-Sampson…