arXiv subjects

Mohameden Ahmedou

Publications and source records attributed to Mohameden Ahmedou.

17 recordsLinked to original sources

arXiv2026

Hopf Reduction and Multiplicity-Resolved Endpoints for the Classical Willmore Flow of Hopf Tori

We establish an exact parametrized reduction of the classical Willmore flow of Hopf tori in the round three-sphere to the spherical elastic flow. In Willmore-flow time the reduction is $(\partial_tγ)^{\perp}=-4\nabla_{L^2}\mathcal E(γ)$; its global reconstruction uses a pullback-circle-bundle normalization and does not require preservation of simplicity along flow lines. Combining the known global subconvergence of the spherical elastic flow with Pozzetta's full convergence theorem then yields full smooth convergence of flow lines, modulo domain diffeomorphisms, for every simply parametrized Hopf torus at arbitrary initial energy. We then obtain quantitative conclusions for the unreparametrized trajectory: finite normal $L^2$-metric length, total-variation convergence of the pullback area measures, bounded-Lipschitz convergence of the image measures, and explicit tail estimates. The density of the full varifold endpoint equals the covering multiplicity of the limiting elastic profile, and Pinkall's marked flat lattices converge. In particular, a simple initial Hopf immersion with energy arbitrarily close to $4π^2$ from above can smoothly converge to some parametrization of the Clifford torus with multiplicity two, although its limiting conformal class in moduli space is exactly the square class.

math.AP↗

arXiv2026

Uniqueness of Finite-Time Varifold Limits for the Möbius-Invariant Willmore Flow

We prove uniqueness of finite-time geometric endpoints for the Möbius-invariant Willmore flow in $\mathbb{S}^3$ under uniform quantitative nonumbilicity. The multiplicity-counting varifolds converge, without reparametrization or Möbius renormalization, to a unique integral two-varifold. An intrinsic transport estimate gives quantitative total-variation convergence of the induced area measures on the fixed domain and bounded-Lipschitz Cauchy control of their pushforwards. Together with Allard compactness and rectifiability, this upgrades subsequential compactness to full-trajectory varifold convergence. The limit has generalized Euclidean mean curvature in $L^2$ with the natural endpoint lower-semicontinuity bound. For finite maximal trajectories with initial energy at most $8π$, Jakob's subsequential alternative becomes sequence independent: the limit is zero, or it has unit density and embedded Lipschitz support of genus zero or one. For Hopf-torus trajectories under the same energy bound, nonumbilicity is automatic and the anchored constant-speed profiles converge weakly in $W^{2,2}$ and strongly in $W^{1,2}$ and $C^{1,α}$ for every $α<\frac{1}{2}$. At infinite time, the same method yields a unique limit under an additional finite-dissipation-length condition.

math.AP↗

arXiv2026

Boundary-weighted barycenter spaces

We study boundary-weighted barycenter spaces, in which an interior support point has cost (or weight) two and a boundary support point has cost one. These spaces are finite-dimensional topological models for the concentration patterns produced by noncompact boundary Euler--Lagrange functionals: an interior bubble carries twice the quantized mass of a boundary bubble, and very negative sublevels are therefore modeled by boundary-weighted rather than ordinary barycenters. The paper gives a systematic algebraic-topological treatment of these spaces. We construct the mixed strata, the closed-stratum poset, and the triangular boundary-weighted colimit filtration; compute the Euler characteristic; and prove a homology decomposition in terms of ordinary barycenter spaces of $\partial M$ and $M/\partial M$. We then specialize the formulas to compact connected orientable surfaces with boundary and to hemispheres, obtaining explicit rational Betti polynomials and, for surfaces, the mod-two polynomials required by the mean-field application. Finally, we explain how these polynomials measure the topology at infinity in the resonant Neumann mean-field equation on a compact surface with boundary.

math.DG↗

arXiv2026

Blow-up solutions for mean field equations with non-quantized singularities on Riemann surfaces with boundary

We study mean field equations with singular sources on a compact Riemann surface with boundary $(Σ,g)$, subject to homogeneous Neumann boundary conditions: \[ -Δ_g v = ρ\left( \frac{V e^{v}}{\int_ΣV e^{v}\, d v_g} - \frac{1}{|Σ|_g}\right) - \sum_{ξ\in Q} \frac{\varrho(ξ)}{2}γ(ξ) \left(δ_ξ- \dfrac{1}{|Σ|_g}\right) \text{in }Σ; \qquad \partial_{ν_g} v = 0 \text{ on }\partialΣ. \] Here, $V$ is a smooth positive function, $ρ$ is a non-negative parameter, $Q\subsetΣ$ is a finite set of prescribed singular points, and the singular weights satisfy $γ(ξ)\in(-1,+\infty)\setminus(\mathbb{N}\cup\{0\})$. The coefficients are given by $\varrho(ξ)=8π$ for $ξ\inΣ\setminus\partialΣ$ and $\varrho(ξ)=4π$ for $ξ\in\partialΣ$. We construct blow-up solutions in the non-quantized singular regime, including purely singular and mixed singular-regular blow-up cases, with parameters approaching resonant values. The construction is achieved via a Lyapunov-Schmidt reduction under suitable stability assumptions. Key words: Singular mean field equations, Blow-up phenomena, Lyapunov-Schmidt reduction, Riemann surfaces with boundary

math.AP↗

arXiv2025

A degree-counting formula for a Keller-Segel equation on a surface with boundary

In this paper, we consider the following Keller-Segel equation on a compact Riemann surface $(Σ, g)$ with smooth boundary $\partialΣ$: \[ -Δ_g u = ρ\Big(\frac{V e^u}{\int_Σ V e^u \mathrm{d} v_g} - \frac{1}{|Σ|_g}\Big) \text{ in } Σ, \quad \text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, \] where $V$ is a smooth positive function on $Σ$ and $ρ> 0$ is a parameter. We perform a refined blow-up analysis of bubbling solutions and establish sharper a priori estimates around their concentration points. We then compute the Morse index of these solutions and use it to derive a counting formula for the Leray-Schauder degree in the non-resonant case (i.e., $ρ\notin 4 π\mathbb{N}$). Our approach follows the strategy suggested by Y. Y. Li [33] and later implemented by C.-S. Lin and C.-C. Chen [15,16] for the mean field equations on closed surfaces and employs techniques from Bahri's critical points at infinity [8].

math.AP↗

arXiv2025

Blow-up solutions for the steady state of the Keller-Segel system on Riemann surfaces

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -Δ_g u +βu =λ\left(\frac{Ve^u}{\int_Σ Ve^u d v_g}-1\right), &\text { in } \mathringΣ\\ \partial_{ ν_g} u=0, &\text { on } \partial Σ\end{array} \right.,\] on a compact Riemann surface $(Σ, g)$ of unit area, with interior $\mathringΣ$ and smooth boundary $\partial Σ$. Here, $Δ_g$ denote the Laplace-Beltrami operator, $dv_g$ the area element of $(Σ, g)$, and $ν_g$ the unit outward normal to $\partial Σ$ and $λ$ and $β$ are non-negative parameters, $V$ is non-negative with finite zero set. For any integers $m>0$ and $k,l\geq 0$ with $m=2k+l$, we establish a sufficient condition on $V$ for the existence of a sequence of blow-up solutions as $λ$ approaches the critical values $4πm$, which blows up at $k$ points in the interior and $l$ points on the boundary. Moreover, the study expands to the corresponding singular problem.

math.AP↗

arXiv2025

Blow-up solutions for mean field equations with Neumann boundary conditions on Riemann surfaces

On a compact Riemann surface $(Σ, g)$ with a smooth boundary $\partial Σ$, we consider the following mean field equations with Neumann boundary conditions: $$ -Δ_g u = λ\left(\frac{Ve^u}{\int_Σ Ve^u \, dv_g} - \frac{1}{|Σ|_g}\right) \text{ in } Σ\text{ with } \partial_{ν_g} u = 0 \text{ on } \partial Σ, $$ We find conditions on the potential function $V: Σ\to \mathbb{R}^+$ such that solutions exist for the parameter $λ$ when it is in a small right (or left) neighborhood of a critical value $4π(m+k)$ for $k \leq m \in \mathbb{N}_+$ and blow up as $λ$ approaches the critical parameter. The blow-up occurs exactly at $k$ points in the interior of $Σ$ and $(m-k)$ points on the boundary $\partial Σ$.

math.AP↗

arXiv2024

Partial Blow-up Phenomena in the $SU(3)$ Toda System on Riemann Surfaces

This work studies the partial blow-up phenomena for the $SU(3)$ Toda system on compact Riemann surfaces with smooth boundary. We consider the following coupled Liouville system with Neumann boundary conditions: $$ -Δ_g u_1 = 2ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac 1 {|Σ|_g}\right) - ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ and $$ -Δ_g u_2 = 2ρ_2\left( \frac{V_2 e^{u_2}}{\int_Σ V_2 e^{u_2} \, dv_g} - \frac{1}{|Σ|_g}\right) - ρ_1\left( \frac{V_1 e^{u_1}}{\int_Σ V_1 e^{u_1} \, dv_g} - \frac{1}{|Σ|_g}\right) \text{in} \,\mathringΣ$$ with boundary conditions $ \partial_{ν_g} u_1 = \partial_{ν_g} u_2 = 0 \text{ on} \, \partial Σ,$ where $(Σ, g)$ is a compact Riemann surface with the interior $\mathringΣ$ and smooth boundary $\partialΣ$, $ρ_i$ is a non-negative parameter and $V_i$ is a smooth positive function for $i=1,2$. We construct a family of blow-up solutions via the Lyapunov-Schmidt reduction and variational methods, wherein one component remains uniformly bounded from above, while the other exhibits partial blow-ups at a prescribed number of points, both in the interior and on the boundary. This construction is based on the existence of a non-degeneracy solution of a so-called shadow system. Moreover, we establish the existence of partial blow-up solutions in three cases: (i) for any $ρ_2>0$ sufficiently small; (ii) for generic $V_1, V_2$ and any $ρ_2\in (0,2π)$; (iii) for generic $V_1, V_2$, the Euler characteristic $χ(Σ)<1$ and any $ρ_2\in (2π,+\infty)\setminus 2π\mathbb{N}_+$.

math.AP↗

arXiv2024

On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting

Given a smooth positive function $K$ on the standard sphere $(\mathbb{S}^n,g_0)$, we use Morse theoretical methods and counting index formulae to prove that, under generic conditions on the function $K$, there are arbitrarily many metrics $g$ conformally equivalent to $g_0$ and whose scalar curvature is given by the function $K$ provided that the function is sufficiently close to the scalar curvature of $g_0$. Our approach leverages a comprehensive characterization of blowing-up solutions of a subcritical approximation, along with various Morse relations involving their indices. Notably, this multiplicity result is achieved without relying on any symmetry or periodicity assumptions about the function $K$.

math.AP↗

arXiv2024

Nirenberg problem on high dimensional spheres: Blow up with residual mass phenomenon

In this paper, we extend the analysis of the subcritical approximation of the Nirenberg problem on spheres recently conducted in \cite{MM19, MM}. Specifically, we delve into the scenario where the sequence of blowing up solutions exhibits a non-zero weak limit, which necessarily constitutes a solution of the Nirenberg problem itself. Our focus lies in providing a comprehensive description of such blowing up solutions, including precise determinations of blow-up points and blow-up rates. Additionally, we compute the topological contribution of these solutions to the difference in topology between the level sets of the associated Euler-Lagrange functional. Such an analysis is intricate due to the potential degeneracy of the involved solutions. We also provide a partial converse, wherein we construct blowing up solutions when the weak limit is non-degenerate.

math.AP↗

arXiv2023

Equilibria of vortex type Hamiltonians on closed surfaces

We prove the existence of critical points of vortex type Hamiltonians \[ H(p_1,\ldots, p_N) = \sum_{{i,j=1},{i\ne j}}^N Γ_iΓ_jG(p_i,p_j)+ψ(p_1,\dots,p_N) \] on a closed Riemannian surface $(Σ,g)$ which is not homeomorphic to the sphere or the projective plane. Here $G$ denotes the Green function of the Laplace-Beltrami operator in $Σ$, $ψ:Σ^N\to\mathbb{R}$ may be any function of class $C^1$, and $Γ_1,\dots,Γ_N\in\mathbb{R}\setminus\{0\}$ are the vorticities. The Kirchhoff-Routh Hamiltonian from fluid dynamics corresponds to $ψ= -\sum_{i=1}^N Γ_i^2h(p_i,p_i)$ where $h:Σ\timesΣ\to\mathbb{R}$ is the regular part of the Laplace-Beltrami operator. We obtain critical points $p=(p_1,\dots,p_N)$ for arbitrary $N$ and vorticities $(Γ_1,\dots,Γ_N)$ in $\mathbb{R}^N\setminus V$ where $V$ is an explicitly given algebraic variety of codimension 1.

math.AP↗

arXiv2022

Non simple blow ups for the Nirenberg problem on half spheres

In this paper we study a Nirenberg type problem on standard half spheres $(\mathbb{S}^n_+,g_0)$ consisting of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary $\partial \mathbb{S}^n_+$. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g_0} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} where $K \in C^2(\mathbb{S}^n_+)$ is a positive function. We construct, under generic conditions on the function $K$, finite energy solutions of a subcritical approximation of $(\mathcal{P})$ on half spheres of dimension $n \geq 5$, which exhibit multiple blow up of \emph{cluster-type} at the same boundary point. These solutions may have zero or non zero weak limit and may develop clusters at different boundary points. Such blow up phenomena on half spheres drastically contrast with the case of the Nirenberg problem on spheres, where non simple blow up for finite energy subsolutions cannot occur and unveils an unexpected connection with vortex type problems arising in Euler equations in fluid dynamic and mean fields type equations in mathematical physics. We construct also, under suitable conditions on the restriction of $K$ on $\partial \mathbb{S}^n_+$, approximate solutions of arbitrarily large energy and Morse index

math.AP↗

arXiv2022

The Nirenberg problem on half spheres: A bubbling off analysis

In this paper we perform a refined blow up analysis of finite energy approximated solutions to a Nirenberg type problem on half spheres. The later consists of prescribing, under minimal boundary conditions, the scalar curvature to be a given function. In particular we give a precise location of blow up points and blow up rates. Such an analysis shows that the blow up picture of the Nirenberg problem on half spheres is far more complicated that in the case of closed spheres. Indeed besides the combination of interior and boundary blow ups, there are non simple blow up points for subcritical solutions having zero or nonzero weak limit. The formation of such non simple blow ups is governed by a vortex problem, unveiling an unexpected connection with Euler equations in fluid dynamic and mean fields type equations in mathematical physics.

math.AP↗

arXiv2022

Classification and a priori estimates for the singular prescribing $Q$-curvature equation on 4-manifold

On $(M,g)$ a compact riemannian $4-$manifold we consider the prescribed $Q-$curvature equation defined on $M$ with finite singular sources. We first prove a classification theorem for singular Liouville equations defined on $\mathbb R^4$ and perform a concentration compactness analysis. Then we derive a quantization result for bubbling solutions and establish a priori estimate under the assumption that certain conformal invariant does not take some quantized values. Furthermore we prove a spherical Harnack inequality around singular sources provided their strength is not an integer. Such an inequality implies that in this case singular sources are \emph{isolated simple blow up points}.

math.AP↗

arXiv2021

The Nirenberg problem on high dimensional half spheres: The effect of pinching conditions

In this paper we study the Nirenberg problem on standard half spheres $(\mathbb{S}^n_+,g), \, n \geq 5$, which consists of finding conformal metrics of prescribed scalar curvature and zero boundary mean curvature on the boundary. This problem amounts to solve the following boundary value problem involving the critical Sobolev exponent: \begin{equation*} (\mathcal{P}) \quad \begin{cases} -\D_{g} u \, + \, \frac{n(n-2)}{4} u \, = K \, u^{\frac{n+2}{n-2}},\, u > 0 & \mbox{in } \mathbb{S}^n_+, \frac{\partial u}{\partial ν}\, =\, 0 & \mbox{on } \partial \mathbb{S}^n_+. \end{cases} \end{equation*} where $K \in C^3(\mathbb{S}^n_+)$ is a positive function. This problem has a variational structure but the related Euler-Lagrange functional $J_K$ lacks compactness. Indeed it admits \emph{critical points at infinity}, which are \emph{limits} of non compact orbits of the (negative) gradient flow. Through the construction of an appropriate \emph{pseudogradient} in the \emph{neighborhood at infinity}, we characterize these \emph{critical points at infinity}, associate to them an index, perform a \emph{Morse type reduction} of the functional $J_K$ in their neighborhood and compute their contribution to the difference of topology between the level sets of $J_K$, hence extending the full Morse theoretical approach to this \emph{non compact variational problem}. Such an approach is used to prove, under various pinching conditions, some existence results for $(\mathcal{P})$ on half spheres of dimension $n \geq 5$.

math.AP↗

arXiv2021

Morse inequalities at infinity for a resonant mean field equation

In this paper we study the following mean field type equation \begin{equation*} (MF) \qquad -\D_g u \, = \varrho ( \frac{K e^{u}}{\int_{\Sig} K e^{u} dV_g} \, - \, 1) \, \mbox{ in } Σ, \end{equation*} where $(Σ, g)$ is a closed oriented surface of unit volume $Vol_g(Σ)$ = 1, $K$ positive smooth function and $\varrho= 8 πm$, $ m \in \N$. Building on the critical points at infinity approach initiated in \cite{ABL17} we develop, under generic condition on the function $K$ and the metric $g$, a full Morse theory by proving Morse inequalities relating the Morse indices of the critical points, the indices of the critical points at infinity, and the Betti numbers of the space of formal barycenters $B_m(Σ)$.\\ We derive from these \emph{Morse inequalities at infinity} various new existence as well as multiplicity results of the mean field equation in the resonant case, i.e. $\varrho \in 8 π\N$.

math.AP↗

arXiv2016

The resonant boundary $Q$-curvature problem and boundary-weighted barycenters

Given a compact four-dimensional Riemannian manifold $(M, g)$ with boundary, we study the problem of existence of Riemannian metrics on $M$ conformal to $g$ with prescribed $Q$-curvature in the interior $\mathring{M}$ of $M$, and zero $T$-curvature and mean curvature on the boundary $\partial M$ of $M$. This geometric problem is equivalent to solving a fourth-order elliptic boundary value problem (BVP) involving the Paneitz operator with boundary conditions of Chang-Qing and Neumann operators. The corresponding BVP has a variational formulation but the corresponding variational problem, in the case under study, is not compact. To overcome such a difficulty we perform a systematic study, à la Bahri, of the so called {\it critical points at infinity}, compute their Morse indices, determine their contribution to the difference of topology between the sublevel sets of associated Euler-Lagrange functional and hence extend the full Morse Theory to this noncompact variational problem. To establish Morse inequalities we were led to investigate from the topological viewpoint the space of boundary-weighted barycenters of the underlying manifold, which arise in the description of the topology of very negative sublevel sets of the related functional. As an application of our approach we derive various existence results and provide a Poincaré-Hopf type criterium for the prescribed $Q$-curvature problem on compact four dimensional Riemannian manifolds with boundary.

math.DG↗