arXiv research

Study on the convergence rate of prescribed scalar curvature flow.

Study of prescribed mean curvature flow on noncompact hypersurfaces in Lorentz manifolds.

In this work, we study the Yamabe flow corresponding to the prescribed scalar curvature problem on compact Riemannian manifolds with negative scalar curvature. The long time existence and convergence of the flow are proved under appropriate conditions on the prescribed scalar curvature function.

Paper solves degenerated circle pattern metric problem in spherical geometry.

Study proves curvature flow existence on CR manifolds.

The Ollivier Ricci flow with prescribed curvature on infinite graphs.

Study of metrics with prescribed curvature and geodesic curvature on a disc.

Mean curvature flow converges to a translating soliton with prescribed contact angle.

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 …

The paper solves curvature problems on graphs using a special flow.

The paper solves curvature problems on infinite hyperbolic surfaces.

The paper studies a flow to prescribe curvature on CR manifolds.

Proves existence and uniqueness of Killing graphs with prescribed curvature.

Study on curvature flow in 4D ball, proving existence and convergence.

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

Unified flow approach to curvature problem on specific manifolds.

Study translating solitons in Minkowski space with prescribed Gauss image.

Paper proves minimizing movements match smooth droplet flow in 3D.

Constructs ancient solutions to mean curvature flow with prescribed singular sets.

Study on prescribing curvature on specific manifolds with negative Gauduchon degree.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

The paper establishes a discrete uniformization theorem for surfaces with piecewise hyperbolic metrics.

We investigate the prescribed Q-curvature flow for GJMS operators with non-trivial kernel on compact manifolds of even dimension. When the total Q-curvature is negative, we identify a conformally invariant condition on the nodal domains of functions in the kernel of the GJMS operator, allowing us to prove the global ex…

We illustrate an example of a generic, positive function K on a Riemannian manifold to be conformally prescribed as the scalar curvature, for which the corresponding Yamabe type L2-gradient flow exhibits non compact flow lines, while a slight modification of it is compact.

Let $(M^{n},g_{0})$ be a $n=3,4,5$ dimensional, closed Riemannian manifold of positive Yamabe invariant. For a smooth function $K>0$ on $M$ we consider a scalar curvature flow, that tends to prescribe $K$ as the scalar curvature of a metric $g$ conformal to $g_{0}$. We show global existence and in case $M$ is not confo…

In this note, we study Q-curvature flow on $S^4$ with indefinite nonlinearity. Our result is that the prescribed Q-curvature problem on $S^4$ has a solution provided the prescribed Q-curvature $f$ has its positive part, which possesses non-degenerate critical points such that $Δ_{S^4} f\not=0$ at the saddle points and …

It is proved the existence and uniqueness of graphs with prescribed mean curvature in Riemannian submersions fibered by flow lines of a vertical Killing vector field.

We employ three different methods to prove the following result on prescribed scalar curvature plus mean curvature problem: Let $(M^n,g_0)$ be a $n$-dimensional smooth compact manifold with boundary, where $n \geq 3$, assume the conformal invariant $Y(M,\partial M)<0$. Given any negative smooth functions $f$ in $M$ and…

The paper constructs surfaces with prescribed mean curvature in a specific space.

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier conditi…

The paper studies Ricci flow on graphs with prescribed curvature.

The paper solves a curvature problem in hyperbolic space using a flow approach.

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 …

The paper studies how certain surfaces evolve in space-time.

We investigate the existence, convergence and uniqueness of modified general curvature flow of convex hypersurfaces in hyperbolic space with a prescribed asymptotic boundary.

Modeling bone microarchitecture adaptation using geometric flows.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

Solves curvature equations using parabolic flows in various spaces.

We define a new version of modified mean curvature flow (MMCF) in hyperbolic space $\mathbb{H}^{n+1}$, which interestingly turns out to be the natural negative $L^2$-gradient flow of the energy functional defined by De Silva and Spruck in \cite{DS09}. We show the existence, uniqueness and convergence of the MMCF of com…

The main purpose of this short note is to point out that the negative gradient flow for the prescribed $\mathbf Q$-curvature problem on $S^n$ can be extended to handle the case that the $\mathbf Q$-curvature candidate $f$ may change signs.

Flow adjusts curvature to avoid a fixed region, proving bounds and regularity.

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is $f$, when $f$ possesses certain reflection or rotation symmetry.

We prove the existence of closed hypersurfaces of prescribed scalar curvature in globally hyperbolic Lorentzian manifolds provided there are barriers.

We show existence of homothetically shrinking solutions of the fractional mean curvature flow, whose boundary consists in a prescribed numbers of concentric spheres. We prove that all these solutions, except from the ball, are dynamically unstable.

Study of Ricci flow convergence on surfaces with boundary.

Paper introduces new flows to find circle packings with specific curvature.

Survey on rigidity results for graphs with prescribed mean curvature.

In this paper we study nonparametric mean curvature type flows in $M\times\mathbb{R}$ which are represented as graphs $(x,u(x,t))$ over a domain in a Riemannian manifold $M$ with prescribed contact angle. The speed of $u$ is the mean curvature speed minus an admissible function $ψ(x,u,Du)$. Long time existence and unif…