New periodic solutions found for a specific equation.
problem Finding solutions to a specific partial differential equation.
method Using Bäcklund transformation for the sine-Gordon equation to generate solutions.
result New periodic exact solutions of the constant astigmatism equation.
We introduce an algebraic formula producing infinitely many exact solutions of the constant astigmatism equation zyy+(1/z)xx+2=0 from a given seed. A construction of corresponding surfaces of constant astigmatism is then a matter of routine. As a special case, we consider multisoliton solutions of t…
In this paper we continue investigation of the constant astigmatism equation z_{yy} + (1/z)_{xx} + 2 = 0. We newly interpret its solutions as describing spherical orthogonal equiareal patterns, with relevance to two-dimensional plasticity. We show how the classical Bianchi superposition principle for the sine-Gordon eq…
We introduce a nonlocal transformation to generate exact solutions of the constant astigmatism equation zyy+(1/z)xx+2=0. The transformation is related to the special case of the famous Bäcklund transformation of the sine-Gordon equation with the Bäcklund parameter λ=±1. It is also a nonlocal symmetry…
The paper classifies and constructs rotational surfaces with constant astigmatism in space forms.
problem Classifying surfaces with constant astigmatism in space forms.
method Classification and construction of surfaces using variational problems and binormal evolution.
result Locally constructed all rotational surfaces of constant astigmatism.
The techniques used in this paper are based on the exterior calculus of Maurer-Cartan forms, and Weingarten surfaces are used to illustrate the methods that apply to quadratic exterior equations with constant coefficients. Isothermic {\it surfaces of constant astigmatism} (non-linear Weingarten surfaces whose differenc…
The paper classifies equations describing spherical or pseudospherical surfaces.
problem Equations describing spherical or pseudospherical surfaces.
method Classification based on compatibility condition of linear problems.
result A complete and explicit classification of equations of the form ztt=A(z,zx,zt)zxx+B(z,zx,zt)zxt+C(z,zx,zt). This paper explores evolving spheres to Hopf spheres using integer coefficients.
problem Conditions for evolving rotationally symmetric spheres to Hopf spheres.
method Integer linear combination of radii of curvature.
result The fate of an initial sphere is determined by the local geometry of umbilic points.
The paper studies non-integer curvature flows and proves convergence to spheres under specific conditions.
problem Analyzing the convergence of non-integer curvature flows on rotationally symmetric surfaces.
method Spectral theory of singular Sturm-Liouville operators to construct an eigenbasis and prove convergence.
result The flow converges to a round sphere if the focal points coincide at the poles, otherwise to a non-round Hopf sphere.
Study of families of lines on spheres and their focal sets.
problem Characterizing families of lines on spheres and their geometric properties.
method Analyzing submanifolds of TSn and their focal sets, using symplectic structures and sectional curvatures. result Derivation of formulas relating sectional curvatures of focal sets to differences in radii of curvature of generating hypersurfaces.
Constructs surfaces with constant mean curvature from Bessel equation.
problem Creating surfaces with constant mean curvature.
method From the Bessel equation, constructs immersions of the twice-punctured Riemann sphere into R^3.
result Family of constant mean curvature surfaces constructed.
Study shows non-uniqueness and failure of compactness in constant curvature equations.
problem Non-uniqueness and failure of compactness in constant curvature equations.
method Warped product manifold construction and smooth counterexample creation.
result Compactness of solutions fails for dimensions ≥ 62.
The paper proves constant rank theorems for special Lagrangian equations.
problem Understanding saddle solutions and Liouville type results for special Lagrangian equations.
method Argument based on saddle solutions and Liouville type results for the special Lagrangian equation.
result Obtained constant rank theorems for saddle solutions to the special Lagrangian equation and the quadratic Hessian equation.
The abstract shows how constant mean curvature surfaces in hyperbolic space are linked to the Liouville equation.
problem The constant mean curvature one equation for surfaces in hyperbolic 3 space.
method Group theoretic constructions and equivalence of quotient representations.
result The Darboux integrability of the equations shows that they admit equivalent quotient representations.
Constructs periodic solutions for wave equations, including Einstein's, with negative cosmological constant.
problem Finding periodic solutions for nonlinear wave equations, especially Einstein's equations with negative cosmological constant.
method Analytic continuation to construct periodic solutions.
result Infinite-dimensional families of time-periodic solutions for vacuum and Einstein-Maxwell-dilaton-scalar fields.
The nonlinear equations describing all the nonsingular pencils of metrics of constant Riemannian curvature are derived and the integrability of these nonlinear equations by the method of inverse scattering problem is proved. It is proved that all the nonsingular pairs of compatible metrics of constant Riemannian curvat…
Linearized Einstein equations simplified via Calabi operator.
problem Linearizing Einstein equations for cosmological applications.
method Using the Calabi operator from projective differential geometry.
result Linearized Einstein equations simplified.
Compactness fails for curvature equations in high dimensions.
problem Compactness of solutions to curvature equations fails in high dimensions.
method Chen and Wu constructed a smooth counterexample.
result Compactness fails in dimensions not less than 35.
Solves geodesic equations on specific metrics types.
problem Finding explicit solutions for geodesic equations on Eguchi-Hanson metrics.
method Analyzes geodesic equations under different constant conditions.
result Explicit solutions not yet available for geodesic equations when only ψ is constant. Study constant Q-curvature metrics on conic 4-manifolds.
problem Find metrics with constant Q-curvature on conic 4-manifolds.
method Analyze related differential equations in the given conformal class.
result Solve constant Q-curvature problem on conic 4-manifolds.
We provide a congruence theorem for minimal surfaces in S5 with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in S5 with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…
Study finds nodal solutions of Yamabe equation constant along isoparametric levels.
problem Existence of nodal solutions for the Yamabe equation.
method Constant along isoparametric levels of a function.
result Existence of nodal solutions proven.
We propose a new two-component geodesic equation with the unusual property that the underlying space has constant positive curvature. In the special case of one space dimension, the equation reduces to the two-component Hunter-Saxton equation.
Proves solvability of general inverse σ_k equations with constant coefficients.
problem Solvability of general inverse σ_k equations with constant coefficients.
method Proves existence of unique solution if a C-subsolution exists.
result Confirms analytical conjecture for deformed Hermitian--Yang--Mills equation.
Paper proves biharmonic hypersurfaces in nonzero space form have constant mean curvature.
problem Proving constant mean curvature for biharmonic hypersurfaces.
method Careful analysis of Gauss and Codazzi equations.
result Positive answer to Balmus-Montaldo-Oniciuc's conjecture for four dimensional hypersurfaces.
Study on surfaces in Heisenberg group with constant mean curvature.
problem Constant mean curvature surfaces in the Heisenberg group.
method Proves surfaces in a neighborhood of non-umbilic points are solutions to a sinh-Gordon equation with a differential constraint.
result Surfaces in Heisenberg group with constant mean curvature described by solutions to sinh-Gordon equation.
Study Dirac-Einstein equations on manifolds with boundary, focusing on constant volume and chiral conditions.
problem Analyzing Dirac-Einstein equations on manifolds with boundary conditions.
method Characterizing bubbling phenomena and classifying ground state bubbles, proving an Aubin-type inequality.
result Proved an Aubin-type inequality and existence result.
We study Einstein's equation in (m+n)D and (1+n)D warped spaces (Mˉ,gˉ) and classify all such spaces satisfying Einstein equations Gˉ=−Λˉgˉ. We show that the warping function not only can determine the cosmological constant Λˉ but also it can determine the cosmological constant Λ a…
The paper studies constant Q-curvature metrics on manifolds.
problem Finding constant Q-curvature metrics on manifolds.
method Global bifurcation theory applied to a fourth-order ordinary differential equation.
result Multiplicity of positive solutions and constant Q-curvature metrics.
We study the wave analog of the Liouville equation and the constant mean curvature equations in 2 space dimensions, which are energy critical. We exhibit a blow-up criteria for the former using tools from conformal geometry, and we exhibit finite time blow-up for the latter under suitable assumptions on the initial dat…
Solves geodesic equations on specific metrics.
problem Explicit geodesic solutions for certain metrics are not known.
method Solves geodesic equations under different constant conditions.
result Explicit solutions not available for all conditions.
Study solutions to metric equations on modified surfaces.
problem Finding constant scalar curvature Kähler metrics on modified surfaces.
method Expanding solutions to extremal metric type equations.
result Developed methods to solve metric equations on modified surfaces.
The study of harmonic diffeomorphisms reduces to solving a specific Beltrami equation.
problem Classifying harmonic diffeomorphisms between surfaces.
method Reduction to solving a Beltrami equation and an elliptic sinh-Gordon equation.
result Solutions to the sinh-Gordon equation classify harmonic maps.
The paper characterizes surfaces in Heisenberg group with constant p-mean curvature.
problem Characterizing surfaces with constant p-mean curvature in the Heisenberg group. method Using the fundamental theorem of surfaces in H1, the existence of constant p-mean curvature surfaces is linked to solutions of a nonlinear ODE. result Complete set of solutions to the ODE (1.2) or (1.5) divides constant p-mean curvature surfaces into several classes. Solves constant pre-factor problem for tt*-Toda equations using asymptotic data and symplectic structures.
problem Constant pre-factor problem for the tt*-Toda equations.
method Explicit evaluation using asymptotic data and introduction of symplectic structures.
result Preservation of symplectic structures by Riemann-Hilbert correspondence for wider class of solutions.
The paper proves solutions for Yamabe equations on manifolds with boundary.
problem Existence and multiplicity of positive solutions for Yamabe equations.
method Use of isoparametric functions to prove existence and multiplicity results.
result Existence and multiplicity results for positive solutions of Yamabe equations.
A new option pricing model handles non-constant risk aversion and transaction costs.
problem Deriving a pricing model for options with varying risk aversion.
method Developed a transformation method to solve the penalized nonlinear PDE and used finite difference discretization.
result Derived bounds on option prices and proposed a numerical scheme.
In this paper we characterize sprays that are metrizable by Finsler functions of constant flag curvature. By solving a particular case of the Finsler metrizability problem we provide the necessary and sufficient conditions that can be used to decide whether or not a given homogeneous system of second order ordinary dif…
This paper contains a classification of all 3-dimensional manifolds with constant scalar curvature S=0 that carry a non-trivial solution of the Einstein-Dirac equation.
Gradient estimates for hyperbolic space CMC equation solved.
problem Gradient estimates for solutions to constant mean curvature equation in hyperbolic space.
method Maximum principles theory of Φ-functions.
result Gradient estimates obtained for bounded strictly convex domains.
Study proves radial symmetry of solutions to certain nonlinear equations in space forms.
problem Proving radial symmetry of solutions to nonlinear equations in space forms.
method Establishing Rellich-Pohožaev type identities for Hessian quotient and k-Hessian equations.
result Radial symmetry of solutions for Hessian quotient and k-Hessian equations in space forms.
Survey on heat equation estimates on manifolds.
problem Estimating heat equations on manifolds.
method Recalling and discussing Li-Yau, Hamilton, Perelman's estimates and their applications.
result Sharp constants and improved curvature conditions for heat equations on manifolds.
Study on self-similar solutions of supercritical Fujita equation, proving entropy and energy gap.
problem Characterization and stability of solutions to supercritical Fujita equation.
method Introduction of F-functional, F-stability, and entropy; use of mean curvature flows. result Constant solution has lowest entropy among bounded positive self-similar solutions.
The study examines symmetry of solutions to a mean field equation on the 2-sphere, leading to rigidity results for Hawking mass.
problem Symmetry of solutions to a specific elliptic equation on the 2-sphere.
method Analysis of the elliptic equation and conditions for constant solutions.
result The equation has only constant solutions under certain conditions, implying rigidity of Hawking mass.
We construct solutions of the constraint equation with non constant mean curvature on an asymptotically hyperbolic manifold by the conformal method. Our approach consists in decreasing a certain exponent appearing in the equations, constructing solutions of these sub-critical equations and then in letting the exponent …
Paper analyzes blowup of regularized Jang solutions and constant expansion surfaces.
problem Blowup behavior of regularized solutions to Jang equation inside apparent horizons.
method Two geometric treatments: dilation and translation. Characterization of limits of rescaled and translated solutions.
result Limits of properly rescaled solutions are constant expansion surfaces.
Study Kähler metrics with constant scalar curvature using coupled equations.
problem Finding Kähler metrics with constant scalar curvature.
method Solving a system of elliptic equations for a Kähler metric and a closed (1,1)-form, proving higher order estimates and smooth convergence.
result Smooth convergence to a cscK metric coupled to a harmonic (1,1)-form under uniform estimates.
All anti-self-dual Einstein metrics with non-zero cosmological constant arise from a single second-order PDE.
problem Anti-self-dual Einstein metrics with non-zero cosmological constant
method A second-order PDE introduced by Lipstein and Nagy
result All such metrics arise from this equation