Papers

We prove the existence of infinitely many lemniscate-like, butterfly-like, and circular-wave critical points for the length-penalised ideal energy. For the lemniscate and butterfly family, we use a staged direct minimisation process to establish existence. A boundary-layer analysis reveals critical points near two Fresnel phases: $3\pi/4$ modulo $2\pi$, corresponding to lemniscates, and $7\pi/4$ modulo $2\pi$, which are the butterflies. The family of circular waves bifurcates, in a sense, from multiply-covered circles. We use an adapted shooting method to establish their existence.

We study the $X$-ADM mass on complete, one-ended asymptotically flat Riemannian manifolds without boundary in arbitrary dimension $n\geqslant 3$. Starting from a weighted gradient–divergence-free decomposition of the vector field $X$, we construct a conformal metric whose scalar curvature is governed by the critical modified scalar curvature $\mathrm{R}_{X}^{(1-n)}$ associated with $X$. This yields an exact decomposition of the $X$-ADM mass into the ADM mass of the conformal metric plus a nonnegative defect term, which vanishes precisely when $X$ is a gradient, thereby giving an alternative decomposition-based proof of positivity with a full rigidity statement. Explicit negative-mass examples show that the range of parameters $k$ in the modified scalar curvature $\mathrm{R}_{X}^{(k)}$ condition is sharp. We then apply the $X$-positive mass theorem to systems of vector fields, obtaining higher-dimensional multi-charge mass inequalities with rigidity and global alignment in the equality case. This yields the classical three-dimensional electric-magnetic inequality and its higher-dimensional analogue. In higher dimensions, the magnetic datum $\beta$ is a $2$-form and does not have a canonical scalar charge on the asymptotically flat end. We prove that a selected tensorial flux vanishes when $d\beta \in L^1$. Under a prescribed critical asymptotic ansatz, this flux defines a non-trivial tensorial magnetic charge and we formulate a conditional reduction to the $X$-positive mass theorem.

We prove that every Steklov isospectral family of compact smooth planar domains is compact in the $C^\infty$ topology, answering an open question of Colbois, Girouard, Gordon, and Sher. The proof has two main parts. First, a trace comparison principle for Dirichlet-to-Neumann operators yields monotonicity of negative Steklov zeta values and compactness within a fixed conformal class for genus-zero flat surfaces. Second, we analyze the normalized Steklov determinant on degenerating hyperbolic surfaces with geodesic boundary. Its asymptotics are expressed in terms of shrinking boundary components and small Neumann and Dirichlet eigenvalues, which in genus zero are compared with weighted graph Laplacians. This rules out degeneration under a bound on the total hyperbolic boundary length. We finally establish this bound for planar domains by geometric non-collapse estimates.

Let $M^4\hookrightarrow\mathbb S^5(1)$ be a closed CMC hypersurface with constant scalar curvature and constant third power sum $f_3=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$. We prove that if $M^4$ has exactly two distinct principal curvatures at some point, then it is isoparametric. More precisely, it is a Clifford torus of the form $\mathbb S^1(r)\times\mathbb S^3(\sqrt{1-r^2})$ or $\mathbb S^2(r)\times\mathbb S^2(\sqrt{1-r^2})$, where $0<r<1$. Under the additional Willmore condition, we obtain a complete classification: every closed CMC Willmore hypersurface $M^4\hookrightarrow\mathbb S^5(1)$ with constant scalar curvature is isoparametric. Consequently, it is congruent to a totally umbilic geodesic sphere, the minimal Clifford torus $\mathbb S^2(1/\sqrt2)\times\mathbb S^2(1/\sqrt2)$, the nonminimal Clifford torus $\mathbb S^1(\sqrt3/2)\times\mathbb S^3(1/2)$, or a Cartan minimal hypersurface. The proofs combine trace-free local tensor identities, an algebraic analysis of the possible principal curvature multiplicities, and a weighted differential $3$-form together with a cut-off argument near the set where principal curvatures coalesce. No sign condition on the scalar curvature is imposed.

Let $M^3\to X^4$ be a complete, connected, two-sided stable minimal immersion. We prove that if the ambient sectional curvature is nonnegative and the ambient scalar curvature has a positive uniform lower bound, then $M$ is totally geodesic and its normal Ricci curvature vanishes. No weak bounded geometry assumption and no upper curvature bound are imposed. We also construct a complete metric of strictly positive sectional curvature on $\mathbb{R}^4$ admitting a complete, embedded, one-ended, nonparabolic, two-sided stable minimal hypersurface diffeomorphic to $\mathbb{R}^3$ which is not totally geodesic. The rigidity proof combines spectral splitting theory, a warped $\mu$-bubble construction, and a harmonic function level set argument. The example is obtained by a compactly supported deformation of an example in \cite{CLS}.

We develop a componentwise geometric and monodromy theory for the one-support generalized Lamé equation on an elliptic curve, with singularities at \(0\) and \(\pm p\). Its log-free curve decomposes canonically into irreducible even and non-even components, the former being governed by elliptic Painlevé~VI. For the non-even component, we construct the hyperelliptic spectral curve, Baker–Akhiezer functions, and addition map, and prove that \[ \deg \sigma_{n,p}^{(1)}=n(n+1). \] After quotienting by the involution \(T\mapsto -T\), we identify the non-even spectral curve with the classical Lamé spectral curve of weight \(n\), compatibly with the addition map and the rational function \(\kappa\). This identification is realized by \[ \widetilde B=T^2-n(n+1)\wp(p), \] and associates every non-even generalized Lamé equation with a unique classical Lamé equation on the same elliptic curve having equivalent period monodromy. Fixing \(\widetilde B\) yields an isomonodromic deformation with \(\tau\) fixed. Together with the Painlevé-VI deformation on the even component, it gives a componentwise interpretation of the collision \(p\to0\), and yields a finite descent on the admissible completely reducible locus. The classical spectral, finite-gap, finite-monodromy, and curvature theories consequently transfer to the non-even component. Finally, within the symmetric family $\left(n_0,n_1,n_2,n_3,\frac12,\frac12\right),$ the one-support case forms an affine genus-zero hierarchy, whereas for $\left(1,1,0,0,\frac12,\frac12\right)$, the non-even normalization is generically elliptic and becomes rational on the discriminant locus, while the full compactified log-free curve retains arithmetic genus two. This first genus jump marks the boundary of the affine theory and motivates a genus-dependent componentwise geometry.

We prove that in the anti-Gaussian space $\bigl(\R^{m+1},\bar g_{\Euc},e^{|x|^2/4}\,dx\bigr)$, $m\ge2$, round spheres centered at the origin are the only closed, connected, two-sided immersed hypersurfaces with constant weighted mean curvature that are stable under weighted-volume-preserving variations.

In this short note, we characterize the equality case of Bray's conjecture recently proved by Jiang-Li-Wang \cite{JLW}.

We prove that $S^2\times S^3$ admits a Riemannian metric with positive sectional curvature. We view it as a principal circle bundle over $S^2\times S^2$. A diagonal Cheeger deformation of the base and a connection whose curvature form vanishes on the remaining flat tori yield a nonnegatively curved connection metric whose zero-curvature planes are the horizontal lifts of the tangent planes to those tori. We then perturb this metric by the real part of a global complex-valued symmetric $2$-tensor. Differentiation along the circle fibers produces a trace-free first variation of the second fundamental form on local horizontal lifts of the flat tori. The Gauss equation converts this into a positive second-order curvature term that dominates as the fibers shrink. A quantitative lower bound for the Hessian in directions normal to the set of zero-curvature planes extends this positivity to nearby planes. The metric and the proof are discovered by the Odin Automatic AI Research Agent.

We resolve a question posed by Harvey and Lawson concerning the uniqueness of continuous viscosity solutions to the Dirichlet problem for the Lagrangian mean curvature equation on the unit ball with continuous boundary data. For every dimension $n\geq 2$, we construct a continuous phase $\psi \colon \overline{B_1}\to(-n\frac{\pi}{2},n\frac{\pi}{2})$ and continuous boundary data $g \colon \partial B_1\to \mathbb{R}$ for which the Dirichlet problem \[ \sum_{i=1}^n \arctan\lambda_i(D^2u)=\psi(x)\quad\text{in }B_1, \qquad u=g\quad\text{on }\partial B_1, \] admits a continuum of distinct continuous viscosity solutions.

Along the prey nullcline $y=g(x)$ of a planar system, the Jacobian entry satisfies $J_{11}=P\,g'$, where $P=-f_{1y}|_{\gamma}$. Under $P>0$ and predator self-damping, the Hopf trace condition forces $g'(x^\ast)>0$: critical points of $g$ are spectral barriers, so the Hopf locus lies on ascending branches. This geometric localization determines where oscillatory instability can occur; we determine how the bifurcation unfolds there. Let $D=f_{1x}+f_{2y}$ and $\tau=D|_{\gamma}$. Using nullcline coordinates $(u=x,v=y-g(x))$ and Hadamard's lemma gives $\dot u=-vH$, $\dot v=W$, with $H(u,0)=P$. The mixed jet of $W$ is determined by the jets of $\tau$ and $\nu=f_2|_{\gamma}$, yielding a closed five-term formula for the first Lyapunov coefficient $\ell_1$ in trace-field form, valid for predator-dependent functional responses. Three distinct reductions of the cubic jet are established. In general planar Hopf bifurcations, eigenvector pairing gives $\partial\ell_1/\partial f_{1yyy}=\partial\ell_1/\partial f_{2xxx}=0$. Straightening the nullcline removes two further slots, while within the Gause class the reparametrization $\varphi\mapsto g$ removes $p'''$. They have different scopes. For predator-dependent responses, comparison with the $y$-affine truncation of the $3$-jet at fixed linear part gives $\ell_1=\ell_1^{\mathrm{aff}}+\Delta$, where $\Delta=\Lambda\bigl(f_{1xyy}+\omega_0^{-2}\langle\partial_x f,\nabla D\rangle f_{1yy}\bigr)$ and $\Lambda>0$. On the Hopf locus, $\langle\partial_x f,\nabla D\rangle=J_{11}\tau'+(\omega_0^2/P)D_y$. Thus the same quantity $J_{11}=Pg'$ that localizes the bifurcation at first order reappears at third order as the weight of the tangential trace derivative. The bridge identity is independent of eigenvector normalization, while $\ell_1,\ell_1^{\mathrm{aff}},\Delta,\Lambda$ scale by $|c|^2$ under $q\mapsto cq$; their signs and ratios are unchanged.

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.

We prove that the torsion three-form in the preferred splitting of a left-invariant generalized nilsoliton on a nilpotent Lie group is harmonic. This gives a positive answer to Question 7.12 proposed by Fusi-Lafuente-Stanfield in [3].

For $\mu>0$, let \[ M(\mu)=\left\{(z,w)\in\mathbb{C}^2 : |z|^2+|w|^{2/\mu}<1\right\} \] be the Thullen domain, and let $g_{\mathrm{CY}}$ denote its complete Cheng–Yau Kähler–Einstein metric, normalized by \[ \operatorname{Ric}(g_{\mathrm{CY}})=-3g_{\mathrm{CY}}. \] We prove that, for every $6/7\leq\mu<1$, a suitable rescaling of $g_{\mathrm{CY}}$ admits a global holomorphic isometric immersion into the infinite-dimensional complex hyperbolic space $\mathbb{CH}^{\infty}$. To the best of our knowledge, these are the first examples of complete nonhomogeneous Kähler–Einstein manifolds admitting such an immersion. By \cite[Lemma~6]{DSIL2012}, the same metrics also admit Kähler immersions into the flat Hilbert space $\ell^2(\mathbb{C})$. Since the Thullen domains considered here are nonhomogeneous, these examples are neither totally geodesic complex hyperbolic spaces nor products of rescaled complex hyperbolic spaces. Consequently, they provide counterexamples to \cite[Conjecture~4.1]{LoiZedda2018}, for both the complex-hyperbolic and flat Hilbert-space alternatives, and also disprove the earlier flat-Hilbert rigidity conjecture formulated in \cite[Remark~10]{LoiZedda2011}. As a further consequence, the flat realization yields complete nonhomogeneous $\eta$-Einstein Sasakian manifolds admitting global Sasakian immersions into the infinite-dimensional Heisenberg space form.

We construct surface diffusion flows $f:\mathbb{T}^2\times[0,T)\to\mathbb{R}^3$ that drive smooth closed embedded tori to pinchoff in finite time. The flow remains embedded for $t\in[0,T)$ and develops a curvature singularity only at a distinguished point $p$ as $t\nearrow T$. Away from $(p,T)$ the flow converges smoothly as $t\nearrow T$. We characterise the singularity profile: If $A(t)$ denotes the radius of the waist, and the surface is given locally near the waist by the rotation of a radial graph $(z,t)\mapsto r(z,t)$, then there exists a constant $\mu>0$ and smooth function $U:\mathbb R\to\mathbb R$ such that \[ A(t)=\{4\mu(T-t)\}^{1/4}(1+o(1)), \qquad A(t)^{-1}r(A(t)\zeta,t)\stackrel{C^\infty_{loc}}{\longrightarrow} U(\zeta). \] Here $U$ is a rigorous realisation of the classical fundamental positive even conical similarity profile first computed numerically by Wong, Miksis, Voorhees and Davis and subsequently analysed by Bernoff, Bertozzi and Witelski.

In this paper, we study Ricci curvature bounds for statistical submersions from a direct Ricci-curvature perspective. Although Chen–Ricci inequalities for statistical submersions have already been established, a complementary lower estimate is needed to obtain a two-sided description of the Ricci curvature. Motivated by this observation, we combine the Ricci curvature relations of a statistical submersion with Hineva's algebraic inequality. We first obtain a Chen–Ricci upper estimate and a Hineva-type lower estimate along the vertical distribution, expressed in terms of the intrinsic Ricci curvature of the fibres and the fundamental tensors $T$ and $T^{*}$. The two estimates together provide upper and lower bounds for the vertical Ricci curvature. We then derive a Chen–Ricci inequality along the horizontal distribution involving the fundamental tensors $A$ and $A^{*}$. By combining the vertical and horizontal curvature relations, we further obtain corresponding Chen–Ricci and Hineva-type estimates for the mixed distribution. The equality conditions are characterized in terms of the components of the fundamental tensors and their duals. Explicit examples are given to illustrate both equality and strict inequality cases. Thus, the paper provides a unified Ricci-curvature approach to statistical submersions and, in particular, gives the first Hineva-type lower Ricci curvature estimates for statistical submersions, leading to two-sided Ricci curvature bounds.

The aim of this paper is to classify some special Riemannian manifolds with cyclic parallel Ricci tensor, i.e. \begin{equation} D_{ijk}=\nabla_{i}R_{jk}+\nabla_{j}R_{ki}+\nabla_{k}R_{ij}=0\nonumber \end{equation} These structures include non-compact gradient shrinking Ricci soliton, compact $(m > 1)$-quasi-Einstein manifolds with boundary and critical spaces. We will construct some integral identities and make use of the curvature conditions reasonably to prove that the Ricci tensor is parallel.

For every integer $m\geq 3$, we construct a smooth Riemannian metric $g$ on a manifold diffeomorphic to $S^m$ such that $\text{sec}_g\geq 0$, every sectional curvature is strictly positive away from a closed nowhere-dense set, and the point locus on which some sectional curvature vanishes has Hausdorff dimension exactly $\frac{1}{2}$. The metric is induced on the boundary of a smooth convex body in $\mathbb{R}^{m+1}$. The essential local model is a convex graph whose Hessian has a one-dimensional kernel precisely on a Cantor subset of one coordinate axis; a regularized maximum then inserts this graph into a round sphere without introducing additional degenerate points. The main content of this paper is generated by ChatGPT 5.6 and verified by the author.

We extend the discrete gauge rigidity of time-scaled intertwining cocycles to operator networks over a connected manifold M and classify the holonomy that the continuous setting admits. For dissipative semigroups $S_x(t)=e^{-tA_x}$ linked by transport operators $K_\gamma S_{\gamma(0)}(t)=S_{\gamma(1)}(\lambda(\gamma)t)K_\gamma$ along paths $\gamma$, we prove that the scaling field carries no monodromy: $\lambda(x,y)=\tau(x)/\tau(y)$ for a continuous positive function $\tau$, unique up to a multiplicative constant, with multiplicativity forced by the intertwining relation rather than assumed. For regular cocycles the transport defines a bounded operator connection on a Hilbert bundle over M; the holonomy of a closed loop is confined to the commutant of the generator and decomposes into spectral sectors, on each of which it is adiabatic (Berry-Wilczek-Zee) transport twisted by an independent commutant-valued sector potential; for flat unitary cocycles the holonomy is classified by unitary representations of $\pi_1(M)$ in the commutant. An explicit rotating network over $S^1$ exhibits holonomy equal to the parity operator while the Berry connection one-form of every level vanishes identically: the monodromy is carried by $\mathbb{Z}_2$ orientation classes of Moebius eigenline bundles.

We construct helicity-conditioned statistical states for a Whitney-discretized electromagnetic field on a closed oriented three-manifold. The simplicial de Rham complex provides exact discrete gauge symmetry, while the Whitney inner product separates exact, harmonic, and coexact sectors. The spatial Abelian Chern–Simons functional is gauge invariant and depends only on the coexact potential. After fixing harmonic modes, we introduce a helicity-biased Gaussian ensemble on the reduced electromagnetic phase space and derive explicit formulas for its admissible parameters, partition function, mean helicity, relative entropy, and Fisher information. The distribution uniquely minimizes relative entropy under a prescribed mean-helicity constraint. Its helicity susceptibility equals the variance of the discrete Chern–Simons functional and controls the local distinguishability of neighboring statistical states. Because magnetic helicity is generally not conserved under unconstrained Maxwell dynamics, these states represent conditioned inference rather than dynamical equilibrium.

Integrability of distributions on $\mathbb{Z}$-graded manifolds is examined. First, the Local Frobenius theorem – the local existence of flat coordinates for an involutive distribution – is proved. Then, two different notions of an integral submanifold are discussed. It is found that, for the stronger of the two notions, the Global Frobenius theorem holds, but as one implication only: involutivity implies integrability. It is then shown that integrability implies a certain weaker version of involutivity. Simple counterexamples of the converse implications are given.

We describe a Lean formalization of Hamilton's 1982 theorem on closed, connected three-manifolds with positive Ricci curvature. The development contains a short-time existence theorem for Ricci flow and substantial geometric-analysis infrastructure: Riemannian tensor calculus, the Levi–Civita connection, Ricci-flow evolution equations, scalar and tensor maximum principles, three-dimensional curvature algebra, preservation of Ricci pinching, and Hamilton's improved pinching estimate. The formalization follows an alternative blow-up route, rather than Hamilton's original normalized-flow proof. Its time-uniform short-time existence, maximal continuation, no-local-collapsing, and Cheeger–Gromov–Hamilton compactness pipelines have been formalized and are included in the artifact, while we give only a brief account of these companion developments and record the interfaces and consequences used by the Hamilton argument; a detailed exposition of their full constructions is deferred to the second author's forthcoming thesis. We interweave representative Lean declarations with their mathematical meaning and record the status and provenance of every major component. All source-level status claims are tied to the source release identified below.

We describe in detail and provide computer code for constructing optimal geometric braids of three strands from algebraic data encoding the braiding pattern. Our optimality criterion uses the known interpretation of braids as homotopy classes (rel endpoints) of paths in $SL_2(\mathbb{R})$ joining the identity $I$ to some $A \in SL_2(\mathbb{Z})$, i.e., elements of the fundamental group of $SL_2(\mathbb{R})/SL_2(\mathbb{Z})$, a quotient equivalent to the unit tangent bundle of the classical modular surface $\mathbb{H}/PSL_2(\mathbb{Z})$. The main technical result finds the length minimizing geodesic in a prescribed homotopy class. From another perspective, of independent interest, this amounts to shooting the shortest geodesic that connects, with a prescribed number of spins en route, two given unit tangent vectors to the Poincaré (half-)plane $\mathbb{H}$. The length is measured by a Riemannian metric from a family of deformed Sasaki metrics, sometimes called Kaluza-Klein metrics, whereby unit tangent vectors can be interpreted as infinitesimal rotors, called spinners, and the ratio of the mass to the moment of inertia is the deformation parameter. At the universal covering level $\widetilde{SL}_2(\mathbb{R})$, the geometry is one of Thurston's eight model 3D geometries. In the vanishing mass limit, it converges to the better understood Carnot-Carathéodory contact geometry, where our geodesic shooting extends known formulas. The finite mass case is more delicate, requiring numerical solution of a targeting equation. The characterization of the length minimizing geodesics (No-multiplicity Theorem) and the resulting targeting equation are the main original contribution. The exposition is complete and multi-pronged, aimed at a broad spectrum of readers. (Numerous figures are the backbone of the narrative and should be viewed in color.)

We prove that stable nonlocal minimal cones in three dimensions are flat, for $s$ sufficiently close to zero. Our approach develops new structural properties of stable $s$-minimal surfaces in every dimension as $s \downarrow 0$, notably a compactness theory and pinching theorems near a half-space.

Inspired by quantum field-theoretic constructions of 4d $\mathcal{N}=1$ quiver gauge theories that flow to superconformal field theories (SCFTs), we construct 7d manifolds of $G_2$-holonomy, which geometrically engineer these quivers in M-theory. Field theoretically, the 4d quivers are obtained by flux torus compactifications of 6d $\mathcal{N}=(1,0)$ SCFTs. The 6d theory compactified on a circle gives rise to a 5d KK-theory, which has a geometric realization as M-theory on a non-compact elliptically fibered Calabi-Yau threefold. Following the field-theoretical prescription, these local geometries are fibered over a circle to realize (topological) $G_2$-holonomy manifolds. We carry this out concretely in the case of the rank 1 E-string theory and its 4d quivers and construct new families of $G_2$-holonomy spaces.

In this work we derive local differential Harnack estimates for positive solutions of filtration equations on complete Riemannian manifolds with a lower Ricci curvature bound by using Nash–Moser iteration. Inspired by earlier work on parabolic frequency functions, we establish some new frequency monotonicity formulas for these equations. The results cover the porous-medium and fast-diffusion equations, as well as several non-power diffusion laws.

Let $f:M^n\looparrowright\Sph^{n+q}(1)$, $n\ge2$ and $q\ge1$, be a closed, connected, non-totally-geodesic minimal immersion with second fundamental form $h$, and put $S=|h|^2$ and $S_*=\max_M S$. If $p\in f(M)$ has multiplicity $m$ and $f^{-1}(p)=\{x_1,\ldots,x_m\}$, then \[ \Vol(M)\ge \left[m+\varepsilon_n\sum_{j=1}^m \left(\frac{S(x_j)}{S_*}\right)^2\right]\Vol(\Sph^n), \] where $[110n(n+2)^2]^{-1}<\varepsilon_n<[104n(n+2)^2]^{-1}$. If the immersion is linearly full, then \[ \frac{\Vol(M)}{\Vol(\Sph^n)} \ge \max\!\left\{1+\varepsilon_n, \frac{4(n+1)^n}{(n+3)^{n+2}}(n+q+1)\right\}. \] Moreover, for every hyperplane $H$ through the origin, each connected component of $M\setminus f^{-1}(H)$ has volume at least $4(n+1)^n(n+3)^{-n-2}\Vol(\Sph^n)$; consequently the number of components is at most $\frac{(n+3)^{n+2}}{4(n+1)^n}\frac{\Vol(M)}{\Vol(\Sph^n)}$.

We prove $C^{k/2,\alpha/2}$ -regularity up to a non-umbilic elliptic complex point for the Bishop family of holomorphic discs with boundary in a $C^{k,\alpha}$ regular real surface. Furthermore, we prove existence and regularity of holomorphic discs near certain complex points of index $\ge 2$. The proof employs a novel blow-up of the real surface which resolves the complex point to a pair of totally real surfaces and leads to a $\mathbb{Z}_2$ -equivariant Riemann-Hilbert problem for holomorphic annuli. The index is computed to be 1 and the problem is shown to be Fredholm-regular.

A compact complex manifold has the bounded mass property if, for one (equivalently, every) Hermitian form $\omega$, the masses $\int_X(\omega+\mathrm{dd}^{\mathrm{c}}\varphi)^n$ are uniformly bounded over all smooth $\varphi$ with $\omega+\mathrm{dd}^{\mathrm{c}}\varphi>0$. We prove that this property fails on the Hopf threefold $(\mathbb C^3\setminus\{0\})/\langle z\mapsto\mathrm{e}^{-1}z\rangle$, answering a question of Boucksom–Guedj–Lu.

Let $M^k\subset \overline{B}_R^N$ be a smooth compact connected orientable free boundary $f_c$-minimal submanifold of the closed Euclidean ball, where $f_c(x)=c|x|^2/2$ and $c\ge 0$. Assume that $cR^2\le k$ and $|A_{x^\perp}|^2\le 1+\frac{1}{k-1}(1-c|x^\perp|^2)^2$, where $A_{x^\perp}(X,Y)=\langle x^\perp,A(X,Y)\rangle$. We prove that $M$ is diffeomorphic either to $D^k$ or to $S^1\times D^{k-1}$; strict pinching yields the disk. The proof uses Hessian convexity of the squared-distance function, a nullity estimate along its minimum set, and a sublevel-set argument. In dimension two and codimension one, the non-disk branch is rotationally symmetric. We also construct a local family of embedded rotational examples for small $c\ge 0$, with the $c=0$ member equal to the critical catenoid.