Curvature Problem Research Articles

Correction: On the Chen–Lin Conjecture for the Prescribed Scalar Curvature Problem

Correction: On the Chen–Lin Conjecture for the Prescribed Scalar Curvature Problem

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

A Prescribed Scalar and Boundary Mean Curvature Problem and the Yamabe Classification on Asymptotically Euclidean Manifolds with Inner Boundary

Existence of positive solutions for one dimensional Minkowski curvature problem with singularity

Existence of positive solutions for one dimensional Minkowski curvature problem with singularity

On the Chen–Lin Conjecture for the Prescribed Scalar Curvature Problem

On the Chen–Lin Conjecture for the Prescribed Scalar Curvature Problem

Global bifurcation of positive solutions of a quasilinear indefinite Neumann problem in some FLRW spacetimes

We are concerned with the global structure of the radial positive solution set of the radially symmetric prescribed curvature problem with the Neumann boundary condition in some Friedmann–Lemaître–Robertson–Walker spacetimes. The proof of our main result is based on bifurcation techniques.

A Generalization of Newton–Maclaurin’s Inequalities

Abstract In this paper, we prove Newton–Maclaurin-type inequalities for functions obtained by linear combination of two neighboring primary symmetry functions, which is a generalization of the classical Newton–Maclaurin inequality. Using these inequalities, we also generalize some inequalities of Lin and Trudinger [18], which are relevant to the study of partial differential equations associated with curvature problems.

The prescribed cross curvature problem on the three-sphere

The paper studies the problem of prescribing positive cross curvature on the three-dimensional sphere. We produce several existence results and an example of non-uniqueness, disproving a conjecture of Hamilton's.

A flow approach to the prescribed Gaussian curvature problem in ℍ𝑛+1

Abstract In this paper, we study the following prescribed Gaussian curvature problem: K = f ~ ⁢ ( θ ) ϕ ⁢ ( ρ ) α − 2 ⁢ ϕ ⁢ ( ρ ) 2 + | ∇ ¯ ⁢ ρ | 2 , K=\frac{\tilde{f}(\theta)}{\phi(\rho)^{\alpha-2}\sqrt{\phi(\rho)^{2}+\lvert\overline{\nabla}\rho\rvert^{2}}}, a generalization of the Alexandrov problem ( α = n + 1 \alpha=n+1 ) in hyperbolic space, where f ~ \tilde{f} is a smooth positive function on S n \mathbb{S}^{n} , 𝜌 is the radial function of the hypersurface, ϕ ⁢ ( ρ ) = sinh ⁡ ρ \phi(\rho)=\sinh\rho and 𝐾 is the Gauss curvature. By a flow approach, we obtain the existence and uniqueness of solutions to the above equations when α ≥ n + 1 \alpha\geq n+1 . Our argument provides a parabolic proof in smooth category for the Alexandrov problem in H n + 1 \mathbb{H}^{n+1} . We also consider the cases 2 < α ≤ n + 1 2<\alpha\leq n+1 under the evenness assumption of f ~ \tilde{f} and prove the existence of solutions to the above equations.

Theoretical and Numerical Constant Mean Curvature Surface and Liquid Entry Pressure Calculations for a Combined Pillar–Pore Structure

Modern microfabrication techniques have led to a growing interest in micropillars and pillar–pore structures. Therefore, in this paper a study of the liquid entry pressure of a hydrophobic pillar–pore structure and the corresponding liquid–gas interface shape for the pressurized liquid is presented. We theoretically analysed the constant mean curvature problem for the rotationally symmetric case and determined an analytical expression for the liquid entry pressure of a hydrophobic pillar–pore structure. Furthermore, the shape of the liquid–gas interface as well as a formula for the location of the minimum were derived. The results are useful for designing geometries with specific properties, such as preventing or facilitating liquid intrusion into rough structures. We compared these results to multiphase lattice Boltzmann simulations where equilibrium contact angles in the range of 157∘ to 102∘ were tested. In our further analysis, we compared theoretical findings from previous works to our lattice Boltzmann simulations. The presented cases can serve as a benchmark for the development and validation of numerical multiphase models.

Infinitely Many Regular Weak Solutions for Odd Symmetric Prescribed Mean Curvature Problems

Infinitely Many Regular Weak Solutions for Odd Symmetric Prescribed Mean Curvature Problems

The holomorphic d-scalar curvature on almost Hermitian manifolds

In this paper, we study the existence of conformal metrics with constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed, connected almost Hermitian manifolds of dimension n ⩾ 6. In addition, we obtain an application and a variational formula for the associated conformal invariant.

Doubling the equatorial for the prescribed scalar curvature problem on {{mathbb {S}}}^N

We consider the prescribed scalar curvature problem on {{mathbb {S}}}^N ΔSNv-N(N-2)2v+K~(y)vN+2N-2=0onSN,v>0inSN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta _{{{\\mathbb {S}}}^N} v-\\frac{N(N-2)}{2} v+{\ ilde{K}}(y) v^{\\frac{N+2}{N-2}}=0 \\quad \ ext{ on } \\ {{\\mathbb {S}}}^N, \\qquad v >0 \\quad {\\quad \\hbox {in } }{{\\mathbb {S}}}^N, \\end{aligned}$$\\end{document}under the assumptions that the scalar curvature {tilde{K}} is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of O(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.

Topological invariants for the scalar curvature problem on manifolds

Topological invariants for the scalar curvature problem on manifolds

Non-degeneracy of the Bubble Solutions for the Fractional Prescribed Curvature Problem and Applications

Non-degeneracy of the Bubble Solutions for the Fractional Prescribed Curvature Problem and Applications

Constant rank theorems for curvature problems via a viscosity approach

An important set of theorems in geometric analysis consists of constant rank theorems for a wide variety of curvature problems. In this paper, for geometric curvature problems in compact and non-compact settings, we provide new proofs which are both elementary and short. Moreover, we employ our method to obtain constant rank theorems for homogeneous and non-homogeneous curvature equations in new geometric settings. One of the essential ingredients for our method is a generalization of a differential inequality in a viscosity sense satisfied by the smallest eigenvalue of a linear map Brendle et al. (Acta Math 219:1–16, 2017) to the one for the subtrace. The viscosity approach provides a concise way to work around the well known technical hurdle that eigenvalues are only Lipschitz in general. This paves the way for a simple induction argument.

Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type

This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem:{−Δu1=K1(y)u2p,y∈RN,−Δu2=K2(y)u1q,y∈RN,u1,u2>0, where N≥5,p,q∈(1,∞) with 1p+1+1q+1=N−2N, K1(y) and K2(y) are positive radial potentials.At first, under suitable conditions on K1,K2 and the certain range of the exponents p,q, we construct an unbounded sequence of non-radial positive vector solutions, whose energy can be made arbitrarily large. Moreover, we prove a type of non-degeneracy result by use of various Pohozaev identities, which is of great interest independently. The indefinite linear operator and strongly coupled nonlinearities make the Hamiltonian-type systems in stark contrast both to the systems of Gradient type and to the single critical elliptic equations in the study of the prescribed curvature problems. It is worth noting that, in higher-dimensional cases (N≥5), there have been no results on the existence of infinitely many bubbling solutions to critical elliptic systems, either of Hamiltonian or Gradient type.

On the prescribed scalar curvature problem with very degenerate prescribed functions

On the prescribed scalar curvature problem with very degenerate prescribed functions

On the prescribed negative Gauss curvature problem for graphs

<p style='text-indent:20px;'>We revisit the problem of prescribing negative Gauss curvature for graphs embedded in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{n+1} $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M2">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula>. The problem reduces to solving a fully nonlinear Monge–Ampère equation that becomes hyperbolic in the case of negative curvature. We show that the linearization around a graph with Lorentzian Hessian can be written as a geometric wave equation for a suitable Lorentzian metric in dimensions <inline-formula><tex-math id="M3">\begin{document}$ n\geq 3 $\end{document}</tex-math></inline-formula>. Using energy estimates for the linearized equation and a version of the Nash–Moser iteration, we show the local solvability for the fully nonlinear equation. Finally, we discuss some obstructions and perspectives on the global problem.</p>

New existence for the higher order prescribed curvature problem

We consider the following prescribed curvature problem involving polyharmonic operator on $ \mathbb{S}^{N}: $ \begin{document}$ \begin{align*} D^{m}u = K(|y|)u^{m^{*}-1}, \quad u > 0 \quad \hbox{in} \quad \mathbb{S}^{N}, \quad u \in H^{m}(\mathbb{S}^{N}), \end{align*} $\end{document} where $ K(|y|) $ is a positive function, $ m^{*} = \frac{2N}{N-2m} $ is the critical exponent of Sobolev embedding, $ N>6m. $ $ D^m $ is the $ 2m $ order differential operator given by$ D^m = \prod\limits_{l = 1}^m (-\Delta_g+\frac{1}{4}(N-2l)(N+2l-2)), $where $ \Delta_g $ is the Laplace-Beltrami operator on $ \mathbb{S}^N, $ and $ \mathbb{S}^N $ is the unit sphere with Riemann metric $ g. $ By applying the non-degenerate result of positive multi-bubbling solutions constructed in [15], we construct new type solutions by using Lyaponov Schmidt reduction arguments combining with the gluing method. One will see that this type of new solution is difficult to be obtained through direct critical theory and reduction argument. We consider the problem from a different point of view and glue $ n $ bubble solution to $ k $ bubble solution and obtain new type multi bubble solution.

Conformally Prescribed Scalar Curvature on Orbifolds

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 ge 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 textrm{U}(2)-invariant Leray–Schauder degree for a family of negative-mass orbifolds found by LeBrun.