Papers

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.

We prove that the spectrum of a smooth closed three-manifold gives a positive lower bound for the length of its shortest nonconstant closed geodesic. The argument uses three curvature heat coefficients and the first positive singularity of the retarded wave trace. The heat coefficients yield a curvature–length dichotomy: sufficiently large maximum curvature forces a closed geodesic shorter than the curvature scale. We then show that a shortest closed geodesic lying below the conjugate radius produces a singularity of the sine trace. Combining these estimates gives a lower bound depending only on the common spectrum. The compactness theorem of Anderson then implies smooth compactness of the full isospectral family modulo diffeomorphisms.

In this work we study the long term behavior of the curve shortening flow of asymptotically conical curves with convex projections in $\mathbb{R}^3$.

We complete, in the Riemannian setting, the classification of isolated singularities of positive solutions to the Yamabe equation for \(3\le n\le 29\), initiated in the flat case by Caffarelli, Gidas, and Spruck in 1989. Precisely, every isolated singularity is either removable or asymptotic to a Fowler solution. The sharpness of this range is established by counterexamples for \(n\ge30\) in our subsequent paper.

We show that a Riemannian metric on $RP^3$ is a surface Zoll metric if and only if its four projective area widths are equal. We also establish a weighted Crofton formula on $RP^3$ under surface Zoll metrics, from which we derive an inequality relating the systole, the volume, and the common value of the projective area widths.

The IIB system is a system of PDEs for an SU(3)–structure and a positive function on a 6-manifold. Its solutions describe conformally balanced pluriclosed Hemitian metrics on complex 3-folds with holomorphically trivial canonical line bundle. In particular, solutions of the IIB system are steady solitons for the generalized Ricci-flow and Bismut–Hermitian–Einstein metrics. In this paper we study cohomogeneity one solutions of the IIB system. We construct 1-parameter families (up to scaling symmetries) of complete non-compact non-Kähler solutions on the smoothing of the conifold with controlled geometry at infinity. The generic member of the family is asymptotically conical with tangent cone at infinity the Calabi–Yau cone metric on the conifold. As a limit of the family we recover an explicit solutions known in the physics literature as the Chamseddine–Volkov/Maldacena–Nuñez solution, which has an exotic asymptotic geometry. We show that there are no complete cohomogeneity one solutions on crepant resolutions of the conifold (or its quotient), but we also establish the existence of an analogous 1-parameter family of forward complete solutions defined on exterior domains of the conifold with prescribed incomplete behaviour along the interior boundary and similar asymptotic behaviour at infinity. As a byproduct of these existence results, we produce infinitely many complete non-compact non-Kähler Bismut–Hermitian–Einstein metrics in complex dimension 3 with full holonomy of the Bismut connection, in contrast to the holonomy reduction forced upon compact examples. We also find infinitely many such complete non-compact full-holonomy non-Kähler examples (with at least two ends) in complex dimension 2 by revisiting a construction due to Callan–Harvey–Strominger in the physics literature.

The axisymmetric profile \(\sin\psi(\rho)=c_0\rho\log(\rho/\rho_B)\) introduced by Naito, Okuda, and Ouyang is an exact solution of the unconstrained Helfrich shape equation for every \(\rho>0\), but its mean curvature diverges logarithmically at the dimple center. A covariant local analysis shows that in Cartesian graph coordinates the immersion is nondegenerate and belongs to \(W^{2,p}\) for every finite \(p\); it is \(C^{1,\alpha}\) for every \(\alpha<1\), but neither \(C^{1,1}\) nor \(C^2\). The conformal factor has a finite nonzero limit, so the point has branch multiplicity one and is not a branch point in the sense used in weak-immersion compactness theory. The Helfrich energy is finite, whereas the complete Euler–Lagrange operator carries the distributional residue \(\E_{\CH}=4\pi c_0\delta_p\) under the convention specified below. Hence the profile is a classical solution on the punctured surface and a finite-energy weak immersion, but it is not a free weak critical point of the unforced Canham–Helfrich functional. Stationarity is restored after adding the corresponding point-force potential or under a pinned-point constraint. Area and volume multipliers cannot cancel the Dirac residue. The apparent sign discrepancy between the shape equation used in the original papers and the modern convention is explained by the opposite definition of mean curvature. These results delimit precisely what can and cannot be inferred from modern existence and regularity theorems for Canham–Helfrich minimizers.

We study homotopy theories of higher groupoids in specified geometries, distinguishing geometric representability from realization and descent. Ordinary smooth and Banach-open sheaves admit full Brown categories of fibrant objects. For ordinary geometric groupoids, a crossing-axes obstruction motivates an empty-compatible incomplete Brown structure. Its split-Banach version requires a fixed chart-compatible plot site with local kernel-product closure. These results include all finite unique-horn bounds and their union. For represented derived models, finite matching and an outer-prism filtration give the structural Brown calculus. Nuiten's finite-geometric realization theorem yields the finite derived-smooth structure and its localization. Enriched realization base change on the original small Kan-enriched site gives the outer-unbounded hypercomplete structure, with positive-degree geometric horn conditions and Reedy fibrancy in every degree. For Banach domains, structured spectra, strict-open gluing and split-open hyperdescent establish geometric finite limits and hypersheaf representability. They give the represented Brown structure, using the same hypercompleted weak-equivalence class at every outer bound. A square-zero test detects nonordinary self-intersections of the chosen Banach charts, including infinite-dimensional charts. For connective associative affines, homotopy split horns, Pridham's finite effectivity theorem and a pointwise diagonal-fibration argument give categories of fibrant objects for his Artin and Deligne–Mumford classes. The associative theory uses quasi-isomorphisms of algebras and realization in presheaves.

We introduce a notion of strict stability for the ADM mass on asymptotically flat manifolds with boundary and establish several of its fundamental properties. In particular, we show that a strictly stable static vacuum metric is a strict local ADM mass minimizer, thus proving a local version of Bartnik's Mass Minimizer Conjecture near Euclidean exterior regions. Our approach also gives a new proof of the existence of static vacuum extensions for nearby Bartnik boundary data on arbitrary exterior regions of Euclidean space.

We define a Bergman barycentric extension of boundary homeomorphisms of the complex unit ball by \[ E_B(f)(z)=\barB(f_*\sigma_z), \] where \(\barB\) is the Busemann barycenter for the Bergman metric and \(\sigma_z\) is the visual, equivalently Poisson–Szegő, measure based at \(z\). We prove well-definedness, prescribed boundary values, full \(\Aut(\B^n)\)-naturality, and interior real-analyticity. For every CR-quasisymmetric boundary homeomorphism, the Bergman barycentric extension is a quasi-isometry of complex hyperbolic space; moreover \(E_B(f^{-1})\) is a coarse inverse of \(E_B(f)\). Under sufficiently small positive CR cross-ratio distortion we obtain a sharper Tukia-type theorem: for every \(M>1\), the extension is a real-analytic diffeomorphism satisfying \[ M^{-1}d_B(x,y)\le d_B(E_B(f)(x),E_B(f)(y))\le M d_B(x,y). \] In contrast, for \(n\ge2\) there are smooth CR-orientation-preserving CR-quasisymmetric boundary diffeomorphisms whose barycentric extensions are non-injective and, after normalization, have singular differential. Thus large-scale quasi-isometric control persists on the full CR-quasisymmetric class, whereas local non-degeneracy requires stronger boundary control.

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations – a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

We study a one-parameter family of SO(4)-invariant pluriclosed Hermitian metrics $g_c$ with vanishing Bismut Ricci form on the affine quadric $Q_3\cong TS^3$, previously described in the physics literature. For every real parameter $c$, we prove local existence and uniqueness of a real-analytic solution to the defining singular initial-value problem, together with positivity of the associated metric near the singular orbit $S^3$. We then establish global existence and completeness for $|c|\leq1$ and finite-time degeneration for $|c|>1$. The complete family exhausts all SO(4)-invariant Bismut Hermitian-Einstein metrics on the affine quadric $Q_3$, and joins Stenzel's Kähler Ricci-flat to a metric that was first considered by Chamseddine-Volkov/Maldacena-Nuñez. We also determine the leading asymptotics and derive an explicit scalar curvature formula, proving strict positivity for the non-Kähler members and identifying the change in asymptotic scalar curvature at the endpoint $c=1$. Finally, we show that all these metrics are not Bismut flat and have full Bismut holonomy SU(3), providing new examples of such manifolds.

We study lattice-counting functions on compact integral-integral affine manifolds with corners, which generalize integral unimodular polytopes. We prove analogues of Ehrhart's theorem and Ehrhart-Macdonald reciprocity in this setting.

Let $M^n\subset\mathbb{S}^{n+1}(1)$ be a closed minimal hypersurface with constant $S=|h|^2$ and constant $f_3=\operatorname{tr} h^3$, where $h$ is the shape operator. We prove that $S>n$ implies $S\geq2n$, and that the equality images are precisely the minimal Cartan isoparametric hypersurfaces with three principal curvatures, which occur only in dimensions $n=3, 6, 12, 24$.

Let $\Omega\subset\mathbb R^3$ be an unbounded domain with smooth boundary and let $\Sigma$ be a complete, noncompact, orientable, immersed free boundary minimal surface in $\Omega$, with compact boundary $\partial \Sigma\subset\partial \Omega$ and finite Morse index. We prove that if the mean curvature of $\partial \Omega$ satisfies $H_{\partial \Omega}\ge 0$ along $\partial \Sigma$ and $H_{\partial\Omega}>0$ at some point of $\partial\Sigma$, then \[ \textrm{Ind}(\Sigma)\ \ge\ \frac13\Bigl(2g+k+2\sum_{j=1}^r(d_j+1)-2\Bigr), \] where $g$ is the genus of $\Sigma$, $k$ is the number of connected components of $\partial\Sigma$, $r$ is the number of ends of $\Sigma$ and $d_1,\dots,d_r$ are their respective multiplicities. When $H_{\partial\Omega}$ is only assumed to be nonnegative we obtain $\textrm{Ind}(\Sigma)\ge(2g+k-2)/3$, with the sharp bound $(2g+k-1)/3$ under a mild condition on the ends. The proofs use the harmonic one-form method of Ros and Chodosh–Máximo, with weighted $L^2$ spaces, adapted to the free boundary setting in the spirit of Ambrozio–Carlotto–Sharp. The main new ingredient is the computation of the dimension of the space of harmonic one-forms on a punctured compact Riemann surface with boundary which are tangential along the boundary and square integrable with respect to a weight. For a class of admissible weights $\rho$, we prove that this dimension is $2g+k-1+2\sum_jN_j-\varepsilon$, where $N_j$ is the maximal order of pole allowed by $\rho$ at the $j$-th puncture and $\varepsilon\in\{0,1\}$. For the weight of Chodosh–Máximo one has $N_j=d_j+1$.

We study graph hypersurfaces associated with higher-dimensional ($n>2$) Cobb–Douglas production functions with variable returns to scale. We prove a rigidity theorem showing that every flat model in this class is necessarily homogeneous. The proof relies on an affine dimension-reduction principle that transfers the higher-dimensional flatness condition to suitable two-dimensional restrictions. We then obtain a complete classification in the non-degree-one case: every such flat model is a power of a positive linear form. For degree one, we give a local structural description in terms of envelopes on the rank-one set, with linearity on open rank-zero regions. These results extend the two-input analysis recently developed in [8] to the higher-dimensional setting, where the degree-one case exhibits a genuinely different behavior.

We study a family of coupled geometric evolution equations describing the deformation of a Riemannian metric and a non-abelian gauge field on closed manifolds, which generalizes the Ricci–Yang–Mills flow. We derive interior curvature estimates, find a preserved integral curvature condition, and discover a scaling invariant monotone functional analogue to Perelman's entropy. In particular, we prove strong dominance of the Riemannian curvature over the gauge curvature at finite-time singularities in all dimensions. We also provide a non-trivial explicit example of a shrinking self-similar solution on a $SU(2)$ bundle over $S^4$.

We establish the extension of Milnor's inequality, relating total curvature with the bridge index, to the $C^1$-closure of a knot class, without any restriction on the self-intersections of the limit curve. Together with recent work of Reiter–von der Mosel, this resolves the circular elastic knot conjecture, first predicted by Gallotti–Pierre-Louis in 2007 and then formulated as a mathematical conjecture by Gerlach–Reiter–von der Mosel in 2017. More precisely, if the bridge and braid indices of a tame knot class coincide, then the multiply covered circle is the unique elastic knot.

In this paper, we prove that every rationally connected projective manifold admits a smooth uniformly RC-positive Hermitian metric on its tangent bundle, answering Yang's question and giving a characterization of rational connectedness by uniform RC-positivity. We find an example to show that RC-positivity alone does not characterize rational connectedness. We also prove that uniform RC-positivity of the tangent bundle is preserved under blow-ups along connected smooth centers on compact complex manifolds. In the non-Kähler setting, we construct such metrics on all Hopf and Kato surfaces and obtain classification results for compact complex surfaces.

We prove that every minimizing $p$-harmonic map from $B^3$ into $\mathbb S^3$ is locally $C^{1,\alpha}$ for some $\alpha\in(0,1)$, for every $p>p_0$, where $p_0=\frac{7-\sqrt{17}}{2}\approx 1.44$. This closes the gap between the previously established regularity ranges $[2,2.642]\cup[2.961,3]$ and, in particular, yields full interior regularity for all $p\ge2$. The result also extends regularity to the subquadratic range $p_0<p<2$.

We prove that a stable noncompact hypersurface with constant anisotropic mean curvature in $\mathbb{R}^{n+1}$ has only one end for $n \le 6$, under a natural ellipticity condition on the anisotropy. This provides an anisotropic counterpart of the one-end theorem for stable constant mean curvature hypersurfaces. We remark that the result is also new in $\mathbb R^7 (n=6)$, in the case of anisotropic minimal hypersurfaces. The proof relies on the existence of a Sobolev-type inequality.

We study positive mass and related Brown–York type inequalities for three-dimensional manifolds modeled on a static space with flat toroidal slices. Using inverse mean curvature flow, we first derive an inequality relating the asymptotic geometry at infinity to the size of an interior singularity. This inequality admits a natural interpretation as a positive mass type theorem. We then combine this global inequality with a Shi–Tam type construction to obtain a Brown–York type inequality for boundaries isometric to flat tori. In contrast to the Schwarzschild setting, our argument does not extend directly to general boundary surfaces, and we identify several obstructions to such a generalization. Finally, we discuss examples illustrating the relationship between negative mass, the strength of interior singularities, and the topology of the interior.

Basing on new curvature conditions for the Bochner technique of Petersen-Wrink in \cite{PW21}, in this paper, we give some vanishing properties for weighted $L^Q$ $p$-harmonic $\ell$-forms on weighted manifolds, where $p>1, Q>p-1.$ These results generalize some previous theorems for $L^Q$ harmonic $\ell$-forms of Dung-Dung-Hung on the paper \cite{DDH25}.

Motivated by problems in instanton continuation, we establish geometro-topological selection rules for continuing a Riemannian region into a Lorentzian region through a smooth transverse change of signature. Given a collar-gluing $M=\overline{M_R} \cup_{\mathcal{H}}\overline{M_L}$ of a smooth manifold $M$ along a hypersurface $\mathcal{H}$, we show that an admissible signature-changing metric of the form $\widetilde g=g-fV^\flat\otimes V^\flat$ with $g$ a Riemannian metric, having Riemannian signature on $M_R$ and Lorentzian signature on $M_L$, exists if and only if a relative Euler class obstruction on $\overline{M_L}$ vanishes. This obstruction combines ordinary Euler characteristics $\chi$ in a manner that depends on which components of $\mathcal{H}$ represent big bangs and which represent big crunches. It also restricts the choice of $M_R$; for example, if $M$ is closed and $\dim M$ is even, then it requires$\chi(\overline{M_R})=\chi(M)$. We also determine and classify the possible compactness types of $\overline{M_R}$, $\mathcal{H}$, and $\overline{M_L}$ occurring in such a collar-gluing construction, and construct explicit signature-changing metrics in each case. Topology change can be entirely contained within the Riemannian region when $M$ is the interior of a compact manifold with boundary.

Given an even-dimensional smooth manifold and a (co)(closed) bundle-valued $n$-form $a$ that "resembles" a scalar $(n,0)$-form, we may (or may not) recover a "$\mathbb{Z}_2$-(almost) complex structure" under certain assumptions. In the case of a $4$-manifold or a quaternion-Kähler manifold, we can recover a compatible $\mathbb{Z}_2$-complex structure on the whole manifold including the zero locus of $a$. In general, we can recover a $\mathbb{Z}_2$-(almost) complex structure wherever $a$ satisfies a certain nondegeneracy condition, but not necessarily on the whole manifold, with a counterexample identified. The above condition of "resembling" a scalar $(2,0)$-form is satisfied by any $\mathrm{U}(1)$-invariant solution to the Vafa-Witten equation on a compact Kähler surface. This hints at a possibility to generalise the notion of $\mathrm{U}(1)$-invariant solutions to oriented Riemannian $4$-manifolds. However, our results above imply any solution satisfying the condition, and thus seemingly any potential "generalised $\mathrm{U}(1)$-invariant solution", must be trivial if the $4$-manifold admits no compatible $\mathbb{Z}_2$-complex structure.

We prove a sharp effective equidistribution theorem for closed geodesics on a compact hyperbolic surface. This is achieved by carefully analyzing the Selberg trace formula twisted by a Maass eigenfunction, which we refer to as Zelditch's trace formula, following Zelditch's observation that the geometric side can be expressed as a sum of period integrals of eigenfunctions on closed geodesics. The main technical input is a sharp upper bound for the matrix coefficients $\langle \phi, \varphi^2 \rangle$ uniformly in $t_\phi\to \infty$ and $t_\varphi \to \infty$, where $t_\phi$ and $t_\varphi$ are the eigenparameters corresponding to Maass forms $\phi$ and $\varphi$ on $X$.

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 $(\partial_t\gamma)^{\perp}=-4\nabla_{L^2}\mathcal E(\gamma)$; 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 $4\pi^2$ 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.

Let $M^n$, $n\ge3$, be a closed oriented minimal hypersurface in the unit sphere with constant scalar curvature and constant $f_3=\tr(h^3)$, where $h$ is the second fundamental form. We prove that $S=\norm h^2>n$ implies $S\ge2n$. Equality holds if and only if each connected component is locally congruent to a Cartan minimal isoparametric hypersurface with three distinct principal curvatures.

Bogomolov–Guan manifolds form the only known series of compact simply connected non-Kähler holomorphic symplectic manifolds. We prove that all Bogomolov–Guan manifolds of complex dimension at least four are nonformal over $\mathbb{Q}$. The proof constructs a uniform degree-eight $a$-Massey product in a finite-dimensional invariant de Rham model of the symmetric quotient of the Bogomolov–Guan precursor. Its indeterminacy vanishes, and its nontriviality is detected by an explicit top-degree pairing arising from an $S_{n+1}$-invariant tensor contraction. A nonzero-degree map then transfers nonformality to the smooth Bogomolov–Guan manifold. We also compare the Bogomolov–Guan precursor with the nilmanifold appearing in Guan's original construction, showing that they are related by a natural $S_{n+1}$-equivariant finite cover of degree $(n+1)^2$ and have the same invariant de Rham model. Using Guan's resolution, which we prove to be projective and semismall, together with Saito's decomposition theorem, we further show that the third Betti number of every Bogomolov–Guan manifold of complex dimension at least four is equal to $3$.

We show that among Möbius bands in $\mathbb{S}^3$ bounded by a great circle, the minimal Willmore energy is realized by an embedded minimal Möbius band with Morse index two. To prove this, we introduce a $2$-parameter “canonical family" associated to any non-orientable surface in $\mathbb{S}^3$ with boundary a great circle and apply a min-max argument. The canonical family detects the Euler number of the surface and is inspired by the $5$-parameter family discovered by F.C. Marques and A. Neves detecting the genus of an orientable surface in $\mathbb{S}^3$. For $\mathbb{Z}_2$-invariant Klein bottles, this reduces R. Kusner's 1989 conjecture that $\tau_{1,2}$ minimizes the Willmore energy for a Klein bottle immersed in $\mathbb{S}^3$ to any of several conjectural characterizations of the Lawson Möbius band.