Study on existence and structure of P-area surfaces in Heisenberg group.
problem Existence and structure of P-area minimizing surfaces in the Heisenberg group.
method Characterization of existence and structure using an underlying vector field N, proving existence even without satisfying boundary conditions, and applying Barrier condition.
result Existence of P-area minimizing surfaces under certain conditions, providing new understanding of the Heisenberg group.
Study finds both existence and non-existence of maps in Morse boundaries.
problem Existence and non-existence of Cannon-Thurston maps in Morse boundaries.
method Examined Morse boundaries for normal subgroups.
result Found both existence and non-existence of Cannon-Thurston maps.
Study on Lane-Emden equation on curved spaces, revealing new existence and non-existence phenomena.
problem Existence and non-existence of positive solutions for the Lane-Emden equation on Riemannian models.
method Analysis of the subcritical Lane-Emden equation on various Riemannian manifolds with polynomial volume growth.
result Subcritical regime divides into three ranges with distinct existence and non-existence phenomena.
The (α,β)-Ricci-Yamabe flow exists on closed manifolds.
problem Existence of solutions to the (α,β)-Ricci-Yamabe flow. method Showed short time existence and established long time existence theorems.
result Existence of smooth solutions to the (α,β)-Ricci-Yamabe flow on closed manifolds. Proves generic existence of spectral networks for many cases.
problem Existence of spectral networks for a broad range of spectral data.
method Generic existence proof for spectral networks.
result Proves existence for a large class of spectral data.
In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time t∈[0,∞) with unif…
The paper studies equations on almost Hermitian manifolds with estimates and existence results.
problem Solving Monge-Ampère type equations on compact almost Hermitian manifolds.
method Derives C∞ a priori estimates and obtains existence results under admissible conditions. result Existence of solutions under admissible conditions for Monge-Ampère type equations.
Study abelianization of Lie algebroids and groupoids, providing conditions for existence.
problem Existence conditions for abelianization of Lie algebroids and groupoids.
method Investigation of abelianization for Lie algebroids and groupoids, providing necessary and sufficient conditions.
result Necessary and sufficient conditions for the existence of abelianization in both Lie algebroids and groupoids.
The paper examines conditions for Einstein multiply warped products and estimates their parameters.
problem Existence and non-existence of non-trivial Einstein multiply warped products.
method Analyzes conditions for the existence or non-existence of Einstein multiply warped products, especially generalized Kasner type.
result Estimates the Einstein parameter that conditions the existence of such metrics.
New methods prove existence of rotating shapes moving in space.
problem Existence of rotating shapes moving in space.
method Different methods to prove existence based on singular ordinary differential equation.
result Existence of rotationally symmetric translating solutions proven without partial differential equations.
A simplified proof of the Alexander-Conway polynomial exists.
problem Existence of the Alexander-Conway polynomial for links in 3D space.
method Presented an accurate detailed exposition of the proof.
result Existence of the Alexander-Conway polynomial proved.
Proves existence of longest paths in sub-Lorentzian problems.
problem Existence of longest paths in sub-Lorentzian problems.
method Generalizes classical theorem for Lorentzian manifolds to sub-Lorentzian problems.
result Longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.
We will discuss existence of center of mass on asymptotically Schwarzschild manifold defined by Huisken-Yau and Corvino-Schoen. Conditions of existence and examples on non existence are given.
In this work, we obtain a existence criteria for the longtime Kähler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of Kähler Einstein metric to the case with possibly unbounded curvature. Moreover, the Kähler Einstein metric with negative scalar curvture must be uniq…
Existence of balanced embedding proved for complex manifold into infinite-dimensional space.
problem Balanced embedding of non-compact complex manifolds into infinite-dimensional projective space.
method Gradient flow in a Hilbert space, long-time existence established by perturbation, convergence depends on a priori bounds.
result Existence of balanced embedding proved in a model case.
Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
No contact Anosov diffeomorphisms exist on odd-dimensional manifolds.
problem Existence of contact Anosov diffeomorphisms on odd-dimensional manifolds.
method Analysis of Anosov diffeomorphisms and invariant contact structures.
result No invariant contact structure exists for Anosov diffeomorphisms on odd-dimensional manifolds.
Study connects curvature bounds to map existence and flow solutions.
problem Existence of lower scalar curvature bounds and measures.
method Relates curvature bounds to map existence and backward limit of Ricci flow solutions.
result Sufficient condition for existence of limiting scalar curvature measure.
The study proves the existence of k-convex hypersurfaces for specific curvature equations.
problem Proving the existence of k-convex hypersurfaces for Hessian curvature equations. method Combining a priori estimates with the continuity method, and establishing a constant rank theorem.
result Existence and uniqueness of k-convex hypersurfaces for both nonhomogeneous and homogeneous Hessian curvature equations. New results on geodesic flows using curve shortening flow.
problem Stability and existence of geodesics in Riemannian surfaces.
method Curve shortening flow approach.
result Existence of infinitely many geodesics intersecting a given one in every primitive class.
This paper establishes geometric obstructions to the existence of complete, properly embedded, mean curvature flow self-translating solitons Σn⊆Rn+1, generalizing previously known non-existence conditions such as cylindrical boundedness.
The heat flow for Dirac-harmonic maps on Riemannian spin manifolds is a modification of the classical heat flow for harmonic maps by coupling it to a spinor. It was introduced by Chen, Jost, Sun, and Zhu as a tool to get a general existence program for Dirac-harmonic maps. For source manifolds with boundary they obtain…
Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformatio…
Existence of solutions to the Heath-Jarrow-Morton equation of the bond market with linear volatility and general Lévy random factor is studied. Conditions for existence and non-existence of solutions in the class of bounded fields are presented. For the existence of solutions the Lévy process should necessarily be with…
The paper investigates polyharmonic helices in 3D solvable Lie group Sol_3 and Euclidean spheres.
problem Existence and classification of polyharmonic helices of order r.
method Analytical and geometric approaches, including Lie group theory and Euclidean sphere analysis.
result Complete classification of proper r-harmonic helices in Sol_3 and new examples in Bianchi-Cartan-Vranceanu spaces.
We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the man…
Study curves evolving by gradient flow of elastic energy, proving existence, smoothing, and convergence.
problem Evolution of curves with fixed length and clamped boundary conditions.
method Negative L2-gradient flow of elastic energy, existence, parabolic smoothing, constrained Lojasiewicz-Simon gradient inequality. result Convergence to a critical point as time tends to infinity.
In this paper, we give the first detailed proof of the short-time existence of Deane Yang's local Ricci flow. Then using the local Ricci flow, we prove short-time existence of the Ricci flow on noncompact manifolds, whose Ricci curvature has global lower bound and sectional curvature has only local average integral bou…
Extends ASD connection existence to 4-manifolds with cylindrical ends.
problem Existence of ASD connections on 4-manifolds with specific ends.
method Extends gluing theorems to cylindrical ends, using mASD connections.
result Establishes existence of ASD connections on 4-manifolds with cylindrical ends.
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
The Poincaré map is widely used to study the qualitative behavior of dynamical systems. For instance, it can be used to describe the existence of periodic solutions. The Poincaré map for dynamical systems with impulse effects was introduced in the last decade and mainly employed to study the existence of limit cycles (…
In the present paper we consider the problem of the existence of pre-semigeodesic coordinates on manifolds with affine connection. We proved that pre-semigeodesic coordinates exist in the case when the components of the affine connection are twice differentiable functions.
Study proves curvature flow existence on CR manifolds.
problem Existence of curvature flow solutions on CR manifolds.
method Prescribed Webster scalar curvature flow approach.
result Proves existence of curvature flow solutions on 3D CR manifolds.
A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…
The paper proves Mabuchi solitons and constants on Fano admissible manifolds.
problem Existence of Mabuchi solitons on Fano admissible manifolds.
method Defined Mabuchi solitons and constants, proved existence and non-existence.
result Fano admissible manifolds admit Mabuchi solitons if and only if the Mabuchi constant is less than 1.
Extends rigidity and existence results for discrete conformal structures on surfaces with boundary.
problem Rigidity and existence of discrete conformal structures on surfaces with boundary.
method Axiomatic framework and classification of discrete conformal structures.
result Extends results by Guo-Luo and Guo to a general context.
The study finds positive Einstein metrics on complex manifolds and spheres.
problem Existence of positive Einstein metrics on complex manifolds and spheres.
method Investigation of cohomogeneity one metrics and use of known Einstein metrics.
result Existence of positive Einstein metrics on S4m+4 and S8. Study local curvature estimates and existence of conformal metrics on noncompact manifolds.
problem Deriving local C0-estimates and existence of conformal metrics with prescribed curvature. method Utilizing Aviles-McOwen's result and its nonlinear extension, combined with asymptotic conditions.
result Proved existence of complete conformal metrics with prescribed curvature functions.
Study of twisted Calabi flow connecting J-flow and Calabi flow on Kähler manifolds.
problem Existence and convergence of twisted Calabi flow on compact Kähler manifolds.
method Analysis of a family of twisted Calabi flows connecting J-flow and Calabi flow, showing long-time existence and convergence to cscK metrics.
result Long-time existence and convergence of twisted Calabi flow to cscK metrics, implying openness of continuity method.
Study finds existence of Q-curvature metrics on even-dimensional manifolds with conical singularities.
problem Existence of Q-curvature metrics on manifolds with conical singularities. method Blow-up analysis of a 2mth-order PDE and variational min-max argument. result First existence result for supercritical conic manifolds (except spheres).
In this paper we prove the existence of coupled Kähler-Einstein metrics on complex manifolds whose canonical bundle is ample. These metrics were introduced and their existence in the said case was proven by Hultgren and Nyström using calculus of variations. We prove the result using the method of continuity. In the pro…
We prove a general existence result for instantaneously complete Ricci flows starting at an arbitrary Riemannian surface which may be incomplete and may have unbounded curvature. We give an explicit formula for the maximal existence time, and describe the asymptotic behaviour in most cases.
Curvature estimates prove existence of smooth hypersurfaces in hyperbolic space.
problem Existence of smooth complete hypersurfaces with constant curvature in hyperbolic space.
method Deriving curvature estimates to prove existence for all curvature values.
result Existence of smooth hypersurfaces for all possible curvature values.
The paper proves short-time existence and uniqueness of Ricci flow on Finsler manifolds.
problem Existence and uniqueness of Ricci flow solutions on Finsler manifolds.
method Investigation of short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds.
result Theorems demonstrating the short-time existence of the flow solution for n-dimensional Finsler manifolds and the uniqueness of the solution for isotropic Finsler manifolds.
Proves existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
problem Existence and uniqueness of solutions to the Lp Gaussian Minkowski problem.
method Analyzes existence and uniqueness of solutions for different values of p.
result Existence and uniqueness of smooth solutions for p > n.
Paper finds a non-existence theorem for certain translators in high dimensions.
problem Non-existence of certain translators in high-dimensional spaces.
method Developed a non-existence theorem and found an example of a translator.
result Non-existence of entire Qn−1-translators in Rn+1. Paper investigates existence of deflators in financial markets.
problem Existence of equivalent local martingale deflators in semimartingale markets.
method Characterization of deflators using modified semimartingale characteristics.
result Existence of deflators can be characterized by modified semimartingale characteristics.
Tian initiated the study of incomplete Kähler-Einstein metrics on quasi-projective varieties with cone-edge type singularities along a divisor, described by the cone-angle 2π(1−α) for α∈(0,1). In this paper we study how the existence of such Kähler-Einstein metrics depends on α. We show that in the negative s…