Researchers prove existence of self-shrinkers for a specific curvature flow.
problem Existence of self-shrinkers for a degree-one curvature flow with a conical end.
method Analyzing a symmetric homogeneous function and a rotationally symmetric cone to prove the existence of self-shrinkers.
result Proved the existence of a self-shrinker asymptotic to a given conical end.
Study shows uniqueness of self-shrinkers for a specific flow.
problem Uniqueness of self-shrinkers for a degree-one curvature flow.
method Analyzes conditions on a function and cone to prove uniqueness.
result Proves at most one self-shrinker exists asymptotic to a given cone.
The paper studies a flow of hypersurfaces preserving mixed volumes and finds convergence to a sphere.
problem Evolution of hypersurfaces under mixed volume preserving flow.
method A flow defined by powers of homogeneous curvature functions of degree one.
result If initial hypersurface satisfies a pinching condition, there exists a unique, smooth solution converging to a round sphere.
Curvature flows in hyperbolic space preserve positive sectional curvature and contract to a point.
problem Preserving positive sectional curvature in contracting curvature flows in hyperbolic space.
method Homogeneous speed flow with positive sectional curvature, including kth mean curvature flow. result Positive sectional curvature is preserved and the hypersurface contracts to a round point in finite time.
The paper studies how the shape of surfaces changes over time using curvature.
problem Understanding how the shape of surfaces evolves over time using curvature.
method The authors use curvature flow with a power of a function of principal curvatures to study the evolution of surfaces.
result The complete smooth strictly convex solution exists and remains a graph until the maximal time of existence.
New pinching estimates control curvature ratios in inverse curvature flows.
problem Controlling curvature ratios in inverse curvature flows.
method Proving pinching estimates for strictly convex hypersurfaces in space forms.
result Smooth convergence of the inverse curvature flow is proven.
The flow preserves curvature and converges to a geodesic sphere in hyperbolic space.
problem Preserving curvature in hyperbolic space.
method Flow of hypersurfaces with specific curvature speed.
result The flow becomes strictly h-convex and converges to a geodesic sphere.
The paper studies a flow of convex hypersurfaces expanding by their support and curvature functions.
problem Analyzing the behavior of expanding hypersurfaces in Euclidean space.
method Introduced a curvature flow with specific speed function and proved the existence and convergence of the flow under certain conditions.
result The flow converges to a round sphere centered at the origin for all time under specific conditions.
We consider the so-called inverse F-curvature flow (IFCF) x˙=−F−1ν in ARW spaces, i.e. in Lorentzian manifolds with a special future singularity. Here, F denotes a curvature function of class (K∗), which is homogenous of degree one, e.g. the n-th root of the Gaussian curvature, and ν the past dire…
We consider embedded hypersurfaces evolving by fully nonlinear flows in which the normal speed of motion is a homogeneous degree one, concave or convex function of the principal curvatures, and prove a non-collapsing estimate: Precisely, the function which gives the curvature of the largest interior sphere touching the…
In this paper, we study entire translating solutions u(x) to a mean curvature flow equation in Minkowski space. We show that if Σ={(x,u(x))∣x∈Rn} is a strictly spacelike hypersurface, then Σ reduces to a strictly convex rank k soliton in Rk,1 (after splitting off trivial factors) wh…
The paper studies how surfaces expand by non-concave curvature functions in 3D spaces.
problem Investigating the expansion of surfaces by non-concave curvature functions in different space forms.
method Flow of convex surfaces expanding by F−α, where F is a smooth, symmetric, increasing, and homogeneous of degree one function of the principal curvatures, and α is a power depending on the space form. result Long time existence and convergence of the flow, and the non-roundness of the limit shape in hyperbolic space.
The paper proves convergence of certain curvature flows to the origin.
problem Analyzing the convergence of specific curvature flows in Euclidean space.
method Examining fully nonlinear contracting curvature flows with given normal speeds.
result The flows converge exponentially to a sphere centered at the origin after rescaling.
The paper studies foliation flows with logarithmic speeds and finds convergence to translating solutions.
problem Flowing foliations with specific curvature speeds and analyzing convergence behavior.
method Analyzes foliations of Rn+1∖{0} with speeds −log(F/f), focusing on uniformly convex hypersurfaces. result There is a distinct leaf MΘ∗ such that flows starting from it converge to a translating solution. Study flows in hyperbolic space to prove curvature inequalities.
problem Prove geometric inequalities for hypersurfaces in hyperbolic space.
method Volume preserving flows and curvature flows for hypersurfaces in hyperbolic space.
result Proves Alexandrov-Fenchel type inequalities for hypersurfaces with positive sectional curvatures.
The paper studies curvature flows in Euclidean and hyperbolic spaces, proving smooth convergence to spheres.
problem Analyzing curvature flows in Euclidean and hyperbolic spaces.
method Introduced a class of expanding flows with specific speed functions and proved their longtime existence and smooth convergence.
result The flows converge smoothly to spheres in Euclidean and hyperbolic spaces under certain conditions.
A convex surface contracting by a strictly monotone, homogeneous degree one function of curvature remains smooth until it contracts to a point in finite time, and is asymptotically spherical in shape. No assumptions are made on the concavity of the speed as a function of principal curvatures.
It was recently proved that embedded solutions of Euclidean hypersurface flows with speeds given by concave (convex), degree one homogeneous functions of the Weingarten map are interior (exterior) non-collapsing. These results were subsequently extended to hypersurface flows in the sphere and hyperbolic space. In the f…
For ordinary knots in R3, there are no degree one Vassiliev invariants. For virtual knots, however, the space of degree one Vassiliev invariants is infinite dimensional. We introduce a sequence of three degree one Vassiliev invariants of virtual knots of increasing strength. We demonstrate that the strongest invariant …
Paper studies inverse curvature flows and solves related geometric problems.
problem Inverse curvature flows and related geometric problems.
method Analyzes a class of expanding flows with specific speeds and proves existence and convergence.
result Proves the existence and convergence of flows under certain conditions, leading to new solutions to geometric problems.
We prove that a primitive harmonic map is equivariant if and only if it admits a holomorphic potential of degree one. We investigate when the equivariant harmonic map is periodic, and as an application discuss constant mean curvature cylinders with screw motion symmetries.
The paper proves a theorem for non-integrable projective distributions of degree one.
problem Classifying non-integrable projective distributions of degree one.
method Proving a singular Darboux type theorem for homogeneous polynomial closed 2-forms of degree one.
result Classification of non-integrable codimension one distributions of degree one.
The study confirms essential self-adjointness for certain differential operators on manifolds.
problem Essential self-adjointness of differential operators on closed manifolds.
method Analyzing the Hamiltonian flow of the symbol of differential operators.
result The conjecture that certain differential operators are essentially self-adjoint if their Hamiltonian flow is complete.
Holomorphic maps of degree one are biholomorphic, confirming a partial order.
problem Understanding the partial order of holomorphic maps.
method Analyzing holomorphic maps of positive degree between compact complex manifolds.
result Holomorphic maps of degree one are biholomorphic.
The paper examines cohomology in degree one for representations of groups under mixing conditions.
problem Implications of non-vanishing cohomology in degree one for group structure.
method Uses harmonic cocycles and properties of subgroups to analyze cohomology.
result FC-centre is finite for mildly mixing unitary representations.
We give a description of degree-one maps between closed, oriented 3-manifolds in terms of surgery. Namely, we show that there is a degree-one map from a closed, oriented 3-manifold M to a closed, oriented 3-manifold N if and only if M can be obtained from N by surgery about a link in N each of whose component…
We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.
As it is well-known, all Vassiliev invariants of degree one of a knot K⊂R3 are trivial. There are nontrivial Vassiliev invariants of degree one, when the ambient space is not R3. Recently, T. Fiedler introduced such invariants of a knot in an R1-fibration over a surface F. They take values in the free…
Triangulations without degree one edges are connected via moves.
problem Connectivity of triangulations without degree one edges.
method 2-3 and 3-2 moves.
result Subgraph of Pachner graph without degree one edges is connected.
The article extends previous work on contracting convex hypersurfaces by nonhomogeneous curvature functions.
problem Contraction of convex hypersurfaces by nonhomogeneous functions of curvature.
method Extending previous results to various cases, showing convergence to asymptotically round points under pinching conditions.
result Convergence to asymptotically round points under suitable rescaling and pinching conditions.
Classifies Killing forms of arbitrary degree on specific nilpotent Lie groups.
problem Classifying Killing forms of arbitrary degree on specific Lie groups.
method Analyzing left-invariant Killing forms on simply connected 2-step nilpotent Lie groups with left-invariant metrics.
result Classified Killing forms when center is at most 2-dimensional.
Finite-type surfaces have a topological Hopf property.
problem Characterizing surfaces with a topological Hopf property.
method Using topological analogs of the Hopf property.
result Infinite-type surfaces do not have the Hopf property.
Associated to a differential character is an integral cohomology class, referred to as the characteristic class, and a closed differential form, referred to as the curvature. The characteristic class and curvature are equal in de Rham cohomology, and this is encoded in a commutative square. In the Hopkins--Singer model…
We study a simple problem that arises from the study of Lorentz surfaces and Anosov flows. For a non decreasing map of degree one h:S1→S1, we are interested in groups of circle diffeomorphisms that act on the complement of the graph of h in S1×S1 by preserving a vo…
Study harmonic function growth on curved spaces, proving inequalities.
problem Understanding growth rates of harmonic functions on curved manifolds.
method Applied a double-sided Price inequality to estimate growth rates.
result Effective estimates for harmonic function growth rates on curved manifolds.
Graph Lie algebras have infinite prolongation if they have a vertex of degree one.
problem Determining when the prolongation of a graph Lie algebra is infinite-dimensional.
method Analyzing labeled direct graphs and their associated Lie algebras.
result Graph Lie algebras are infinite-dimensional if and only if they have a vertex of degree one.
Study shows Heegaard genus relation in 3-manifold amalgamation.
problem Understanding Heegaard genus in 3-manifold amalgamation.
method Examined amalgamation of 3-manifolds, used degree-one maps, and fixed boundary conditions.
result Proved g(M)≥g(N), showing Heegaard genus relation. Alternative proof classifies Kleinian groups with two parabolics.
problem Classifying Kleinian groups with two parabolic generators.
method Alternative proof using characterizations of epimorphisms and degree one maps.
result Complete characterizations of epimorphisms and degree one maps.
New insights into possible Gauss curvatures on S2.
problem Understanding possible Gauss curvatures of metrics on S2. method Employing the smooth Cheeger-Gromov compactness theorem to obtain a priori estimates and proving a proper Fredholm map with well-defined degree.
result Existence and non-existence results for Gauss curvatures in stable regions.
Lower bound for energy of vector fields on hypersurfaces, related to Gauss map degree.
problem Finding a lower bound for the energy of unit vector fields on closed hypersurfaces.
method Introducing functionals and proving minimization for Hopf flows.
result Lower bound for energy depends on the degree of the Gauss map.
Let F′,F be any two closed orientable surfaces of genus g′>g≥1, and f:F→F be any pseudo-Anosov map. Then we can "extend" f to be a pseudo-Anosov map f′:F′→F′ so that there is a fiber preserving degree one map M(F′,f′)→M(F,f) between the hyperbolic surface bundles. Moreover the extension f′ can…
We compute the space of L2 harmonic forms (outside the middle degrees) on negatively curved Kaehler manifolds of finite volume.
Proves uniqueness of geometric flow in various Riemannian manifolds.
problem Proving uniqueness of geometric flow in general Riemannian manifolds.
method Two backward uniqueness theorems for extrinsic geometric flow.
result Backward uniqueness of extrinsic geometric flow in general ambient manifolds.
In this work we characterize branch data of branched coverings of even degree over the projective plane which are realizable by indecomposable branched coverings.
Investigate scalar curvature under geometric flows
problem Behavior of scalar curvature under geometric flows
method Three specific cases: Ricci flow, Kähler-Ricci flow, Laplacian flow
result Long-time existence of flows
Ancient curve shortening flows have entropy and curvature bounds equivalent.
problem Bounding entropy and total curvature for ancient curve shortening flows.
method Equivalence of entropy and total curvature conditions for ancient curve shortening flows.
result Entropy and total curvature bounds are equivalent for ancient curve shortening flows.
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time existence.
Paper proves uniqueness theorems for non-compact mean curvature flow.
problem Proving uniqueness for non-compact mean curvature flow.
method Energy argument and similar method for Ricci flows.
result Generalizes results by Chen and Yin on mean curvature flow with unbounded curvatures.