Yuki Tsukamoto
Prescribed mean curvature problems on the torus has been considered in one dimension. In this paper, we prove the existence of a graph on the $n$-dimensional torus $\mathbb {T}^n$, the mean curvature vector of which equals the normal component of a given vector field satisfying suitable conditions for a Sobolev norm, the integrated value, and monotonicity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper, we establish the curvature estimates for $p$-convex hypersurfaces in $\mathbb{R}^{n+1}$ of prescribed curvature with $p\geq \frac{n}{2}$. The existence of a star-shaped hypersurface of prescribed curvature is obtained. We also prove a type of interior $C^2$ estimates for solutions to the Dirichlet problem of the corresponding equation.
In this paper, we study the Dirichlet problem for $p$-convex hypersurfaces with prescribed curvature. We prove that there exists a graphic hypersurface satisfying the prescribed curvature equation with homogeneous boundary condition. An interior curvature estimate is also obtained.
Fully nonlinear prescribed curvature problems on closed manifolds with negative curvature
本文研究了闭合负曲率流形上基于修改Schouten张量的非线性给定曲率问题，证明在满足特定结构条件时，此类问题总能被解决。
We study the existence of starshaped compact hypersurfaces with prescribed m-th mean curvature in hyperbolic space.
Utilizing a weight matrix we study surfaces of prescribed weighted mean curvature which yield a natural generalisation to critical points of anisotropic surface energies. We first derive a differential equation for the normal of immersions with prescribed weighted mean curvature, generalising a result of Clarenz and von der Mosel. Next we study graphs of prescribed weighted mean curvature, for which a quasilinear elliptic equation is proved. Using this equation, we can show height and boundary gradient estimates. Finally, we solve the Dirichlet problem for graphs of prescribed weighted mean cu…
The problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider prescribed curvature measure problem in hyperbolic space. We obtain the existence of star-shaped k-convex bodies with prescribed (n-k)-th curvature measures (k<n) by establishing crucial C^2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.
The present paper discusses that a prescribed Gauss-Kronecker curvature problem on the product of unit spheres.
We give a classification of non-removable isolated singularities for real analytic solutions of the prescribed mean curvature equation in Minkowski $3$-space.
In this article we study a class of prescribed curvature problems on complete noncompact Riemannian manifolds. To be precise, we derive local $C^0$-estimate under an asymptotic condition which is in effect optimal, and prove the existence of complete conformal metrics with prescribed curvature functions. A key ingredient of our strategy is Aviles-McOwen's result or its fully nonlinear version on the existence of complete conformal metrics with prescribed curvature functions on manifolds with boundary.
In the spirit of the paper "Large conformal metrics of prescribed Gauss curvature on surfaces of higher genus" by Borer-Galimberti-Struwe, where we dealt with the case of a closed Riemann surface $(M,g_0)$ of genus greater than one, here we study the behaviour of the conformal metrics $g_λ$ of prescribed Gauss curvature $K_{g_λ} = f_0 + λ$ on the torus, when the parameter $λ$ tends to one of the boundary points of the interval of existence of $g_λ$, and we characterize their "bubbling behaviour".
We prove an existence theorem for convex hypersurfaces of prescribed Gauss curvature in the complement of a compact set in Euclidean space which are close to a cone.
We consider the problem of finding a metric in a given conformal class with prescribed nonpositive scalar curvature and nonpositive boundary mean curvature on a compact manifold with boundary, and establish a necessary and sufficient condition in terms of a conformal invariant that measures the zero set of the target curvatures.
Prescribed $L_p$ curvature problem
本文研究了规定 $L_p$ 曲率问题，证明了该问题存在光滑、关于原点对称且严格凸的解。
Existence of convex body with prescribed generalized curvature measures is discussed, this result is obtained by making use of Guan-Li-Li's innovative techniques. In surprise, that methods has also brought us to promote Ivochkina's $C^2$ estimates for prescribed curvature equation in \cite{I1, I}.
The gluing technique is used to construct hypersurfaces in Euclidean space having approximately constant prescribed mean curvature. These surfaces are perturbations of unions of finitely many spheres of the same radius assembled end-to-end along a line segment. The condition on the existence of these hypersurfaces is the vanishing of the sum of certain integral moments of the spheres with respect the prescribed mean curvature function.
Planar loops with prescribed curvature via Hardy's inequality
本文研究具预定曲率的平面闭曲线存在性，提出基于变分法的方法，利用Hardy型不等式及相关函数空间进行证明。
This paper is devoted to $C^2$ a priori estimates for strictly locally convex radial graphs with prescribed Weingarten curvature and boundary in space forms. By constructing two-step continuity process and applying degree theory arguments, existence results in space forms are established for prescribed Gauss curvature equation under the assumption of a strictly locally convex subsolution.
In this paper, we consider the problem of prescribing scalar curvature on n-sphere. Assume that the candidate curvature function $f$, which is allowed to change sign, satisfies some kind of Morse index or symmetry condition. By studying the well-known scalar curvature flow, we are able to prove that the flow converges to a metric with the prescribed function $f$ as its scalar curvature.
We consider graphs Sigma^n in R^m with prescribed mean curvature and flat normal bundle. Using techniques of Schoen, Simon and Yau, and Ecker-Huisken, we derive an interior curvature estimate of the form |A|^2<=C/R^2 up to dimension n<=5, where C is a constant depending on natural geometric data of Sigma^n only. This generalizes previous results of Smoczyk, Wang and Xin, and Wang for minimal graphs with flat normal bundle.
We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.
We consider a kind of Yamabe problem whose scalar curvature vanishes in the unit ball $\mathbb{B}^n$ and on the boundary $\mathbb{S}^{n-1}$ the mean curvature is prescribed. By combining critical points at infinity approach with Morse theory we obtain new existence results in higher dimensional case $n\geq 5$, under suitable pinching conditions on the mean curvature function.
We consider the principal eigenvalue problem for the Laplace-Beltrami operator on the upper half of a topological torus under the Dirichlet boundary condition. We present a construction of the upper half of a topological torus that admits the principal eigenfunction having exact numbers of critical points. Furthermore, we manage to identify the locations of all the critical points of the principal eigenfunction explicitly.
We consider the corresponding Christoffel-Minkowski problem for curvature measures. The existence of star-shaped $(n-k)$-convex bodies with prescribed $k$-th curvature measures ($k>0$) has been a longstanding problem. This is settled in this paper through the establishment of a crucial $C^2$ a priori estimate for the corresponding curvature equation on $\mathbb S^n$.
The constant mean curvature hypersurfaces with prescribed gradient image
该文研究具指定梯度像的常平均曲率超曲面的存在性，在ℝⁿ中一致凸有界光滑域的条件下，证明常平均曲率方程第二边值问题存在唯一凸解。
We prove existence and nonexistence results for annular type parametric surfaces with prescribed, almost constant mean curvature, characterized as normal graphs of compact portions of unduloids or nodoids in $\mathbb{R}^{3}$, and whose boundary consists of two coaxial circles of the same radius.
We are concerned with spacelike convex hypersurfaces of positive constant (K-hypersurfaces) or prescribed Gauss curvature in Minkowski space. Our main purpose is to study entire solutions as well as the Dirichlet problem in bounded domains of the related Monge-Ampere equation.
We present three ways to establish general stability inequalities for various classes of 2-immersions in Euclidean spaces of higher codimension
Obstructions to prescribed Q-curvature of complete conformal metrics on $\mathbb{R}^n$
本文研究了 $\mathbb{R}^n$ 上完备共形度量的预定 Q-曲率问题，给出了一个 Bonnet-Mayer 型障碍定理，并揭示了预定函数衰减速率带来的其他障碍。
In the previous paper, it has been proved that the generalized rotational hypersurfaces of O(n-1)-type and O (l+1) x O(m+1)-type, for which the mean curvature is any prescribed continuous function. This paper is a sequel, and a similar existence result is shown for any type.
We prove the existence and uniqueness of radial graphs over a given domain of $\mathbb{S}^{n}$ having boundary on the sphere $\mathbb{S}^{n}$ and whose mean curvature at every point equals a prescribed positive function satisfying suitable barrier-type and monotonicity conditions.
Existence and uniqueness of $t$-graphs of prescribed mean curvature in Heisenberg groups
本文研究任意维Riemannian Heisenberg群中$t$-图的给定平均曲率方程，刻画无Dirichlet边界数据时经典解的存在唯一性条件，并推广至非常数平均曲率及次Riemannian情形。
Fully nonlinear elliptic equations for some prescribed curvature problems on Hermitian manifolds
该研究利用爆破论证与部分一致椭圆性研究Hermitian流形上的完全非线性椭圆方程，将其应用于寻找指定Chern-Ricci曲率的共形Hermitian度量的几何问题，且假设因几何函数理论阻碍几乎达最优。
We prove the existence of metrics with prescribed $Q$-curvature under natural assumptions on the sign of the prescribing function and the background metric. In the dimension four case, we also obtain existence results for curvature forms requiring only restrictions on the Euler characteristic. Moreover, we derive a prescription result for open submanifolds which allow us to conclude that any smooth function on $\mathbb{R}^n$ can be realized as the $Q$-curvature of a Riemannian metric.
An eigenvalue problem for prescribed curvature equations
本文研究星形k-凸闭超曲面的预定σ_k-曲率方程特征值问题，在偶数据条件下证明了唯一特征值及唯一相关超曲面的存在性，并证明该超曲面严格凸，关键方法是对L_p型方程建立关于p的一致估计。
Let $(M, g)$ be a compact Riemannian manifold of dimension $N \geq 5$ and $Q_g$ be its $Q$ curvature. The prescribed $Q$ curvature problem is concerned with finding metric of constant $Q$ curvature in the conformal class of $g$. This amounts to finding a positive solution to \[ P_g (u)= c u^{\frac{N+4}{N-4}}, u>0 {on} M\] where $P_g$ is the Paneitz operator. We show that for dimensions $N \geq 25$, the set of all positive solutions to the prescribed $Q$ curvature problem is {\em non-compact}.
We study the Dirichlet problem of the Abreu equation. The solutions provide the Kahler metrics of constant scalar curvature on the complex torus.
Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in $\mathbb R^3$
该研究证明了三维空间中带预定接触角液滴的最小化运动解与光滑平均曲率流等价，为两类平均曲率运动描述建立了一致性。
In this work we study solutions of the prescribed mean curvature equation over a general domain that do not necessarily attain the given boundary data. To such a solution, we can naturally associate a current with support in the closed cylinder above the domain and with boundary given by the prescribed boundary data and which inherits a natural minimizing property. Our main result is that its support is a $C^{1,α}$ manifold-with-boundary, with boundary equal to the prescribed boundary data, provided that both the initial domain and the prescribed boundary data are of class $C^{1,α}$.
In this note, we prove that the abstract gradient flow introduced by Baird-Fardoun-Regbaoui \cite{BFR}is well-posed on a closed Riemann surface with conical singularity. Long time existence and convergence of the flow are proved under certain assumptions. As an application, the prescribed Gaussian curvature problem is solved when the singular Euler characteristic of the conical surface is non-positive.
In [7], Guan, Ren and Wang obtained a $C^2$ a priori estimate for admissible 2-convex hypersurfaces satisfying the Weingarten curvature equation $σ_2(κ(X))=f(X, ν(X)).$ In this note, we give a simpler proof of this result, and extend it to space forms.
We study and solve the Dirichlet problem for graphs of prescribed mean curvature in $\mathbb R^{n+1}$ over general domains $Ω$ without requiring a mean convexity assumption. By using pieces of nodoids as barriers we first give sufficient conditions for the solvability in case of zero boundary values. Applying a result by Schulz and Williams we can then also solve the Dirichlet problem for boundary values satisfying a Lipschitz condition.
We obtain $C^2$ a priori estimates for solutions of the nonlinear second-order elliptic equation related to the geometric problem of finding a strictly locally convex hypersurface with prescribed curvature and boundary in a space form. Under the assumption of a strictly locally convex subsolution, we establish existence results by using degree theory arguments.
Convex capillary hypersurfaces of prescribed curvature problem
本文将闭凸超曲面的规定$k$-阶Weingarten曲率问题扩展到毛细情形，通过将其转化为球冠上带Robin边界条件的Hessian商方程可解性问题，在自然充分条件下证明了严格凸毛细超曲面的存在性，扩展了毛细Minkowski问题的相关研究。
Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation
本文研究Einstein流形上预定Q-曲率方程的最优分割问题，利用余一维群作用精确描述解域，证明对称解存在性，并给出Paneitz-Branson算子及Yamabe问题变号解的多重性结果。
We consider the problem of finding complete conformal metrics with prescribed curvature functions of the Einstein tensor and of more general modified Schouten tensors. To achieve this, we reveal an algebraic structure of a wide class of fully nonlinear equations. Our method is appropriate and delicate as shown by a topological obstruction. Finally, we discuss Hessian equations and Weingarten equations by confirming a key assumption.
We study the prescribed scalar curvature problem in a conformal class on orbifolds with isolated singularities. We prove a compactness theorem in dimension $4$, and an existence theorem which holds in dimensions $n \geq 4$. This problem is more subtle than the manifold case since the positive mass theorem does not hold for ALE metrics in general. We also determine the $\rm{U}(2)$-invariant Leray-Schauder degree for a family of negative-mass orbifolds found by LeBrun.
In this paper we are concerned with the problem of finding hypersurfaces of constant curvature and prescribed boundary in the Euclidean space, using the theory of fully nonlinear elliptic equations. We prove that if the given data admits a suitable radial graph as a subsolution, then we can find a radial graph with constant curvature and that realizes the prescribed boundary. As an application we prove that if $Ω\subset\mathbb{S}^n$ is a mean convex domain whose closure is contained in an open hemisphere of $\mathbb{S}^n$ then, for $0<R<n(n-1),$ there exists a radial graph of constant scalar c…
In this paper, we consider prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in $\mathbb{H}^{n+1}$. Under some sufficient condition, we obtain an existence result by the standard degree theory based on the a prior estimates for the solutions to the equations. Different from the prescribed Weingarten curvature problem in space forms, we do not impose a sign condition for radial derivative of the functions in the right-hand side of the equations to prove the existence due to the horo-covexity of hypersurfaces in $\mathbb{H}^{n+1}$.
We consider a $C^{1}$ smooth surface with prescribed $p$(or $H$)-mean curvature in the 3-dimensional Heisenberg group. Assuming only the prescribed $p$-mean curvature $H\in C^{0},$ we show that any characteristic curve is $C^{2}$ smooth and its (line) curvature equals $-H$ in the nonsingular domain$.$ By introducing characteristic coordinates and invoking the jump formulas along characteristic curves, we can prove that the Legendrian (or horizontal) normal gains one more derivative. Therefore the seed curves are $C^{2}$ smooth. We also obtain the uniqueness of characteristic and seed curves pa…
The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry
该研究针对近厄米流形的指定Gauduchon标量曲率问题，通过共形变化推导将其转化为带指数非线性的半线性偏微分方程，用上下解方法证明解的存在性与Gauduchon度相关常数的符号有关，负号时给出充要条件，还复现了三类相关Yamabe问题。
We study the spectrum of phase transitions with prescribed mean curvature in Riemannian manifolds. These phase transitions are solutions to an inhomogeneous semilinear elliptic PDE that give rise to diffuse objects (varifolds) that limit to hypersurfaces, possibly with singularities, whose mean curvature is determined by the "prescribed mean curvature" function and the limiting multiplicity. We establish upper bounds for the eigenvalues of the diffuse problem, as well as the more subtle lower bounds when the diffuse problem converges with multiplicity one. For the latter, we also establish asy…
A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient
本文针对椭圆型Weingarten曲率商方程的严格凸解，借助方程算子的凹性不等式与新颖辅助函数，通过初等最大值原理证明其先验内部曲率界，还得到低维特殊拉格朗日曲率方程严格凸解的对应估计。
Let $(X, g^+)$ be an asymptotically hyperbolic manifold and $(M, [\hat{h}])$ its conformal infinity. Our primary aim in this paper is to introduce the prescribed fractional scalar curvature problem on $M$ and provide solutions under various geometric conditions on $X$ and $M$. We also obtain the existence results for the fractional Yamabe problem in the endpoint case, e.g., $n = 3$, $γ= 1/2$ and $M$ is non-umbilic, etc. Every solution we find turns out to be smooth on $M$.
We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized "weighted" eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also s…
Regularity for the prescribed vertical mean curvature system in the plane
本文证明了平面中指定垂直平均曲率系统的Lipschitz解在全域正则，余维数任意且平均曲率在$L^r(r>2)$中，加强并推广了先前关于Lawson-Osserman猜想的结果。
The horospherical $p$-Christoffel-Minkowski and prescribed $p$-shifted Weingarten curvature problems in hyperbolic space
本文研究双曲空间中凸超曲面的对应几何问题，通过建立新满秩定理证明极限球面p-Christoffel-Minkowski问题存在光滑原点对称严格凸解，并提出给定p-偏移Weingarten曲率问题且证明其存在性结果。
We construct a sequence of compact, oriented, embedded, two-dimensional surfaces of genus one into Euclidean 3-space with prescribed, almost constant, mean curvature of the form $H(X)=1+{A}{|X|^{-γ}}$ for $|X|$ large, when $A<0$ and $γ\in(0,2)$. Such surfaces are close to sections of unduloids with small necksize, folded along circumferences centered at the origin and with larger and larger radii. The construction involves a deep study of the corresponding Jacobi operators, an application of the Lyapunov-Schmidt reduction method and some variational argument.
This paper is devoted to the existence of contact forms of prescribed Webster scalar curvature on a $3-$dimensional CR compact manifold locally conformally CR equivalent to the unit sphere $\mathbb{S}^{3}$ of $\mathbb{C}^{2}$. Due to Kazdan-Warner type obstructions, conditions on the function $H$ to be realized as a Webster scalar curvature have to be given. We prove new existence results based on a new type of Euler-Hopf type formula. Our argument gives an upper bound on the Morse index of the obtained solution. We also give a lower bound on the number of conformal contact forms having the sa…
We study the prescribed scalar curvature problem, namely finding which function can be obtained as the scalar curvature of a metric in a given conformal class. We deal with the case of asymptotically hyperbolic manifolds and restrict ourselves to non positive prescribed scalar curvature. Following earlier results, we obtain a necessary and sufficient condition on the zero set of the prescribed scalar curvature so that the problem admits a (unique) solution.
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric $ g $ for both closed manifolds and compact manifolds with boundary, including the interesting cases $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, in dimensions $ n \geqslant 3 $, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is a positive consta…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。