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Miaomiao Zhu

Publications and source records attributed to Miaomiao Zhu.

At least 19 recordsLinked to original sources
arXiv2026

Existence of disks with prescribed mean curvature and contact angle in arbitrary codimension

Under suitable relative homotopy and boundary admissibility assumptions, we establish a min-max theory in arbitrary codimension that produces nonconstant branched immersed disks or spheres in closed Riemannian manifolds, with controlled Morse index, where the prescribed mean curvature type tensor and the nonorthogonal, nonconstant contact angle condition of disks along a supporting submanifold are determined by the same differential 2-form. In Euclidean space, under suitable topological and convex barrier assumptions, we obtain such disks with free boundary on a closed supporting hypersurface and a Morse index bound depending only on the ambient dimension.

math.DG↗
arXiv2026

Energy quantization for Dirac systems over non-collapsed degenerating Einstein manifolds

We study energy quantization for a class of Dirac systems on compact spin Einstein manifolds of dimension \(n\). For a sequence of solutions to a nonlinear Dirac system with uniformly bounded energy on a fixed spin Riemannian manifold, we first establish an energy identity theorem. We then investigate the more complicated case of underlying domain manifolds being a sequence of non-collapsed degenerating spin Einstein manifolds. At an orbifold singular point, three types of bubble spinors can possibly appear, living respectively on \(\mathbb{R}^n\), on a Ricci-flat ALE bubble space, and on the flat cone \(\mathbb{R}^n/Γ\). By developing asymptotic analysis for solutions over degenerating neck regions, we establish that energy identity holds.

math.AP↗
arXiv2026

Asymptotic analysis for approximate harmonic maps from degenerating cylinders and applications to minimal surfaces

We investigate the blow-up analysis and quantitative behavior for a sequence of maps $\{u_n\}_{n=1}^\infty$ from degenerating tori $(T^2,g_n)$ or from degenerating cylinders $(S^1\times [0,π],g_n)$ with free boundary conditions $u_n(S^1\times \{0,π\})\subset K$ to a compact Riemannian manifold $(N,h)$ satisfying $$E(u_n)+\|τ(u_n,g_n)\|_{L^2}\leq Λ<\infty,$$ where $τ(u_n,g_n)$ is the tension field of $u_n$, $K\subset N$ is a smooth submanifold. We establish generalized energy identities and prove that away from bubbles, the asymptotic limit of the necks are either some geodesics on $N$ or some geodesic-like curves on $K$ where some length formulas are given. This partially confirms a conjecture by Ding-Li-Liu \cite{Ding-Li-Liu} in the sense of approximate sequence case. Moreover, we study an evolution system to seek minimal cylinders in a compact Riemannian manifold with free boundary and with arbitrary codimensions. By studying the convergence of the flow at infinity, we obtain some existence results of minimal cylinders with free boundary. Compared with the closed case in, an interesting new phenomenon here is that the neck may converges to a geodesic-like curve on $K$.

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↗
arXiv2026

Blow-up Solutions for General Toda Systems on Riemann Surfaces

In this paper, we study general Toda systems with homogeneous Neumann boundary conditions on Riemann surfaces. Assuming the surface satisfies the ``$k$-symmetric'' condition, we construct a family of bubbling solutions using singular perturbation methods, where the concentration rates of different components occur in distinct orders. In particular, we establish the existence of asymmetric blow-up solutions for the $SU(3)$ Toda system. Furthermore, the blow-up points are precisely located at the ``$k$-symmetric'' centers of the surface. Keywords: Toda system, Neumann boundary condition, Blow-up solutions, $k$-symmetry, Finite-dimensional reduction

math.AP↗
arXiv2026

Parallel mean curvature surfaces with constant contact angle along free boundaries

We classify branched immersed disks in space forms with non-zero parallel mean curvature vector and non-orthogonal constant contact angle along the boundary in 4-dimensional space form. For higher codimensional case, we prove a codimension reduction theorem for branched immersed bordered Riemann surfaces of higher genus with multiple boundary components under the same parallel mean curvature and constant contact angle assumptions. Furthermore, we construct a family of explicit examples of branched minimal immersions satisfying the non-orthonormal constant contact angle free boundary condition, which demonstrate the sharpness of both the classification result and the codimension reduction result.

math.DG↗
arXiv2025

Yang-Mills energy quantization over non-collapsed degenerating Einstein manifolds and applications

We investigate a sequence of Yang-Mills connections $A_j$ lying in vector bundles $E_j$ over non-collapsed degenerating closed Einstein 4-manifolds $(M_j, g_ j)$ with uniformly bounded Einstein constants and bounded diameters. We establish a compactness theory modular three types of bubbles. As applications, we get some quantization results for several important topological number associated with the vector bundles, for instance, the first Pontrjagin numbers $p_1(E)$ of vector bundles over Einstein 4-manifolds and the Euler numbers $χ(M;E)$ of holomorphic vector bundles over Kähler-Einstein surfaces. Furthermore, we get some quantization results about the volume $v(L_j)$ and certain cohomological numbers (e.g. $dim H^0(M_j;L_j)$) of holomorphic line bundles $L_j$ over non-collapsed degenerating Kähler-Einstein surfaces $(M_j,J_j,g_j)$ with the aid of the classical vanishing theorems, the classical Hirzebruch-Riemann-Roch type theorems, and the profound convergence theory of Kähler-Einstein manifolds. In particular, we obtain some interesting identities involving non-collapsed degenerating compact Kähler-Einstein surfaces with non-zero scalar curvature, which indicate that we can know the Euler number of $M_j$ for large $j$ provided some topological information of the limit orbifold $M_\infty$. For Kähler-Einstein Del Pezzo surfaces, an interesting implication is that we can provide some preliminary estimates for the number of singularities of various types in $M_\infty$ in an effective way. As an unexpected surprise, we find an identity which connects Milnor numbers for singularities in $M_\infty$ and the correction terms in the Hirzebruch-Riemann-Roch theorem for orbifolds. Some quantization results can be extended to the case of higher dimensional $n$-manifolds.

math.DG↗
arXiv2025

The qualitative behavior for biharmonic functions on open manifolds

For a complete noncompact Riemannian manifold with nonnegative Ricci curvature, we show that bounded biharmonic functions are constant and the space consists of biharmonic functions with polynomial growth of a fixed rate is finite dimensional. Also, we derive a Weyl type bound for this space. Finally, we present a finite dimensional result for a class of fourth-order operators on $\mathbb{R}^n$ satisfying certain coefficient conditions.

math.DG↗
arXiv2025

The quantitative behavior of $α$-Yang-Mills-Higgs fields on surfaces

We investigate the blow-up behavior of $α$-Yang--Mills--Higgs ($α$-YMH) fields over closed Riemannian surfaces with the target fiber $F = S^{K-1} \subset \mathbb{R}^K$ being the round sphere, focusing on the establishment of the $α$-energy identity and the no-neck property during the bubbling process. A central innovation is the identification of a hidden Jacobian structure through Hodge decomposition and a new conservation law. Furthermore, we derive a Pohozaev-type identity for $α$-YMH fields, which enables refined control of the energy density. Together, these advances ensure the validity of the $α$-energy identity as $α\to 1$. Our analysis further sharpens the Lorentz space estimates from the $L^{2,\infty}$ to the optimal $L^{2,1}$ scale, ultimately yielding the no-neck property in the blow-up regime. These results provide a unified and quantitative framework for understanding singularity formation in variational gauge theories.

math.DG↗
arXiv2024

Min-max theory and existence of H-spheres with arbitrary codimensions

We demonstrate the existence of branched immersed 2-spheres with prescribed mean curvature, with controlled Morse index and with arbitrary codimensions in closed Riemannian manifold $N$ admitting finite fundamental group, where $π_k(N) \neq 0$ and $k \geq 2$, for certain generic choice of prescribed mean curvature vector. Moreover, we enhance this existence result to encompass all possible choices of prescribed mean curvatures under certain Ricci curvature condition on $N$ when $\dim{N} = 3$. When $\dim{N} \geq 4$, we establish a Morse index lower bound while $N$ satisfies some isotropic curvature condition. As a consequence, we can leverage latter strengthened result to construct 2-spheres with parallel mean curvature when $N$ has positive isotropic curvature and $\dim{N} \geq 4$. At last, we partially resolve the homotopy problem concerning the existence of a representative surface with prescribed mean curvature type vector field in some given homotopy classes.

math.DG↗
arXiv2024

Active-RIS-Aided Covert Communications in NOMA-Inspired ISAC Wireless Systems

Non-orthogonal multiple access (NOMA)-inspired integrated sensing and communication (ISAC) facilitates spectrum sharing for radar sensing and NOMA communications, whereas facing privacy and security challenges due to open wireless propagation. In this paper, active reconfigurable intelligent surface (RIS) is employed to aid covert communications in NOMA-inspired ISAC wireless system with the aim of maximizing the covert rate. Specifically, a dual-function base-station (BS) transmits the superposition signal to sense multiple targets, while achieving covert and reliable communications for a pair of NOMA covert and public users, respectively, in the presence of a warden. Two superposition transmission schemes, namely, the transmissions with dedicated sensing signal (w-DSS) and without dedicated sensing signal (w/o-DSS), are respectively considered in the formulations of the joint transmission and reflection beamforming optimization problems. Numerical results demonstrate that active-RIS-aided NOMA-ISAC system outperforms the passive-RIS-aided and without-RIS counterparts in terms of covert rate and trade-off between covert communication and sensing performance metrics. Finally, the w/o-DSS scheme, which omits the dedicated sensing signal, achieves a higher covert rate than the w-DSS scheme by allocating more transmit power for the covert transmissions, while preserving a comparable multi-target sensing performance.

cs.IT↗
arXiv2022

Geometric analysis of the Yang-Mills-Higgs-Dirac model

The harmonic sections of the Kaluza-Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza-Klein model with the Yang-Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler-Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.

math-ph↗
arXiv2021

Quantization for biharmonic maps from non-collapsed degenerating Einstein 4-manifolds

For a sequence of extrinsic or intrinsic biharmonic maps $u_j: M_j\rightarrow N$ from a sequence of non-collapsed degenerating closed Einstein 4-manifolds $(M_j,g_j)$ with bounded Einstein constants, bounded diameters and bounded $L^2$ curvature energy into a compact Riemannian manifold $(N,h)$ with uniformly bounded biharmonic energy, we establish a compactness theory modular finitely many bubbles, which are finite energy biharmonic maps from $\mathbb{R}^4$, or from $\mathbb{R}^4 / Γ$ for some nontrivial finite group $Γ\subset SO(4)$, or from some complete, noncompact, Ricci flat, non-flat ALE 4-manifold (orbifold). To achieve this, we develop a sophisticated asymptotic analysis for solutions over degenerating neck regions.

math.DG↗
arXiv2020

The boundary value problem for Yang--Mills--Higgs fields

We show the existence of Yang--Mills--Higgs (YMH) fields over a Riemann surface with boundary where a free boundary condition is imposed on the section and a Neumann boundary condition on the connection. In technical terms, we study the convergence and blow-up behavior of a sequence of Sacks-Uhlenbeck type $α$-YMH fields as $α\to 1$. For $α>1$, each $α$-YMH field is shown to be smooth up to the boundary under some gauge transformation. This is achieved by showing a regularity theorem for more general coupled systems, which extends the classical results of Ladyzhenskaya-Ural'ceva and Morrey.

math.DG↗
arXiv2019

Energy quantization for a singular super-Liouville boundary value problem

In this paper, we develop the blow-up analysis and establish the energy quantization for solutions to super-Liouville type equations on Riemann surfaces with conical singularities at the boundary. In other problems in geometric analysis, the blow-up analysis usually strongly utilizes conformal invariance, which yields a Noether current from which strong estimates can be derived. Here, however, the conical singularities destroy conformal invariance. Therefore, we develop another, more general, method that uses the vanishing of the Pohozaev constant for such solutions to deduce the removability of boundary singularities.

math.DG↗
arXiv2019

Harmonic maps with free boundary from degenerating bordered Riemann surfaces

We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars and horizontal boundary collars, we establish a generalized energy identity.

math.DG↗
arXiv2018

Energy identity for a class of approximate Dirac-harmonic maps from surfaces with boundary

For a sequence of coupled fields $\{(ϕ_n,ψ_n)\}$ from a compact Riemann surface $M$ with smooth boundary to a general compact Riemannian manifold with uniformly bounded energy and satisfying the Dirac-harmonic system up to some uniformly controlled error terms, we show that the energy identity holds during a blow-up process near the boundary. As an application to the heat flow of Dirac-harmonic maps from surfaces with boundary, when such a flow blows up at infinite time, we obtain an energy identity.

math.DG↗