Papers

In this paper, we study the deformed Hermitian-Yang-Mills (dHYM) equation on the non-compact Calabi-Yau manifold $Z = {\rm{Tot}}({\bf{K}}_{X})$, where $X$ is a rational homogeneous variety. Since $X$ is a Fano variety, $Z$ is a resolution of the singularity at the vertex of the affine cone ${\rm{Aff}}(X)$, provided the cone is built using the anticanonical polarization ${\bf{L}} = {\bf{K}}_{X}^{-1}$. Using the cohomogeneity-one symmetry of the Ricci-flat Kähler metric obtained via the Calabi ansatz on $Z$, we reduce the fully nonlinear PDE underlying the dHYM equation to a scalar, asymptotically autonomous ordinary differential equation (ODE). From this, we determine the exact condition on the topological phase that guarantees global existence of solutions. As an application, we show that every holomorphic line bundle over $Z$ admits a smooth, globally defined Hermitian connection solving the dHYM equation, provided the total phase lies in an explicit open interval determined by the Lie-theoretic data. Also, we prove a rigidity result classifying the exact geometric conditions under which the dHYM solution collapses into a classical Hermitian-Yang-Mills (HYM) connection. The results established generalize previous constructions and provide a substantial new class of examples. Furthermore, the approach presented allows one to study the behavior of the dHYM solutions through ODE methods. Using this approach, we construct the first explicit non-trivial example of a Hermitian-Einstein connection on a line bundle over a non-toric Calabi-Yau manifold which is not dHYM.

Let $(M^n,g)$ be a complete, connected, noncompact Riemannian manifold, and let $f\ge0$ be a proper $C^2$ weight. We assume polynomial bounds on $f$, $|\nabla f|$, and $\Delta f$ (Assumptions H1-H3), with growth exponent $\alpha>0$. We study the drift Laplacian $L_f=\Delta-\langle\nabla f,\nabla\cdot\rangle$ on $L^2(M,e^{-f}dV)$. We assume that its eigenfunctions have finite polynomial growth orders $\gamma_k$, measured on the level sets of $f^{1/\alpha}$. We also assume that $\gamma_k\to\infty$ in spectral order (Assumption H4). We compare this growth filtration with the filtration by eigenvalue $\lambda_k$. For $\alpha>1$, we study the relation $\lambda_k\asymp\gamma_k^{(2\alpha-2)/\alpha}$, called compatibility. An explicit rotationally symmetric example shows that this relation can fail for the full spectrum, even when $\alpha=2$. We prove discreteness and weighted Agmon estimates for $\alpha>1$. A moment estimate bounds the concentration radius in terms of the growth order and a finite-scale prefactor. A lower localization condition gives one direction of the spectral comparison. For exact warped products with a one-dimensional base, we impose the radial growth condition and a regularity assumption on the radial drift. These give $\alpha=2$ and $\lambda_k\asymp\gamma_k$ in the radial sector. For general $(M,g)$, we assume a transitive isometric symmetry of the level sets instead of a warped-product structure. The invariant-sector growth condition and radial-drift regularity then give $\alpha=2$ and $\lambda_k\asymp\gamma_k\asymp k$. Finally, exact polynomiality of the invariant eigenfunctions implies $\alpha=2$ without the drift regularity assumptions. The reduced equation is then Hermite or generalized Laguerre after normalization. The arguments use neither curvature bounds nor soliton equations.

We prove that the Futaki invariant introduced by Futaki, Hattori, and Ornea vanishes identically on every compact locally conformally Kähler manifold with potential, thereby answering a question raised by Ornea and Verbitsky.

We present a gluing construction for a class of 6th order geometric non-linear PDE's on a Kähler manifold or orbifold, arising in the study of Bismut Hermitian Einstein complex $3$-folds, the supersymmetric ${\rm AdS}_3 \times Y_7$ and ${\rm AdS}_2 \times Y_9$ solutions of type IIB and $D=11$ supergravity models in physics, and in Dervan's Z-critical Kähler metrics. As applications, we obtain a desingularization theorem for orbifolds with crepant singularities admitting a solution of the PDE, as well as a blowing-up of points theorem for smooth complex $3$-folds admitting such solutions. These results yield new examples of regular Bismut Hermitian Einstein structures on the smooth manifolds $3(S^2\times S^4)\sharp 4(S^3\times S^3)$, $5(S^2\times S^4)\sharp 6(S^3\times S^3)$ and $S^1\times 9(S^2\times S^3)$, as well as infinite topological types of smooth compact $7$-dimensional manifolds $Y_7$, providing new solutions to the supersymmetric ${\rm AdS}_3 \times Y_7$ type IIB and ${\rm AdS}_2 \times T^2 \times Y^7$ type $D=11$ supergravity models.

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases, such as weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz and trans-Lorentz manifolds. %Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Here we study pseudo-Riemannian geodesic orbit nilmanifolds of metric index three. Those are the geodesic orbit pseudo-Riemannian manifolds $M = G/H$ of signature $(n-3,3)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\g = \h \oplus \n$ (vector space direct sum) with $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian, $2$-step or $4$-step nilpotent. In contrast to the Riemannian, Lorentzian, and trans-Lorentz cases, the two-step conclusion therefore fails, but $3$-step nilpotency remains impossible. The $4$-step case is confined to a Lorentzian derived algebra with an orthogonal complement of index two, and its structure forces an invariant totally isotropic two-plane in that complement. An explicit nine-dimensional example of signature $(6,3)$ proves sharpness. When the metric is degenerate on $[\n,\n]$ we prove the existence of an invariant isotropic subspace that centralizes its orthogonal complement, yielding a double-extension reduction to a geodesic orbit metric nilpotent Lie algebra of strictly smaller index. An eight-dimensional example shows that this relative centrality need not imply centrality in the full Lie algebra.

Let $\mathbf{C}=\partial E\subset\mathbb{R}^{n+1}$ be an area-minimizing hypercone. The cone may have nonisolated singularities. We prove that there is a constant $c_{\mathbf{C}}>0$ such that \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_{\mathbf{C}} \left(\frac{|F\triangle E|}{R^{n+1}}\right)^2 \] for every $R>0$ and every set $F$ of locally finite perimeter with $F\triangle E\Subset B_R$. This extends the unweighted quantitative inequality from regular area-minimizing hypercones to general area-minimizing hypercones. We follow the calibration argument in the author's earlier work on regular cones. The main change is the construction of the vector fields. The pointwise asymptotic estimates for positive Jacobi fields used in the regular case do not directly apply here. We use the minimal foliations constructed by Zhihan Wang on the two sides of the cone. On each leaf, we integrate projection kernels with a positive Jacobi field as the weight. The kernel construction gives the required divergence bound. Wang's weak Harnack inequality and growth estimates give the integral bounds needed to prove linear growth of the vector fields.

We prove that the cone $C(\Sigma)$ in $\mathbb R^4$ over a suitable minimal Möbius band $\Sigma\subset \mathbb S^3$ is locally mass-minimising with coefficients in $\mathbb Z_2$. This proves a conjecture by Brian White in the context of models of boundary singularities of Plateau's problem in ambient dimension $4$ or higher. The key insight is that the cylindrical slices $C(\Sigma)\cap \{(z,w)\in \mathbb R^2 \times \mathbb R^2 : |w|=r\}$ are minimal surfaces in a flat rank-one translation quotient of $\mathbb R^3$. In fact, they correspond to non-orientable minimising helicoids previously studied by Antonio Ros. We also show that, up to rigid motions, the minimising cones $C(\Sigma)\times \mathbb R^{n-4}\subset \mathbb R^n$ are the only non-planar minimal hypercones with boundary $\mathbb R^{n-2}$ which are smooth away from the boundary that satisfy the minimal cylindrical slice property.

We establish an exact composition principle for spherical immersions coupled through coefficient profiles and norm-preserving bilinear maps. Under an explicit mixed-orthogonality condition, the spherical mean-curvature vector splits into orthogonal input and profile contributions. The composition is minimal if and only if every input is minimal and the profile is $W$-minimal for an explicit monomial weight. Using Hsiang–Lawson reduction and Kapouleas–McGrath gluing, we construct closed embedded $W$-minimal profiles, including families with unbounded intermediate Betti numbers. For sufficiently large comparable input dimensions, a Gaussian transfer developed in this paper produces interior profiles from closed embedded Gaussian seeds that are nondegenerate modulo rotations.

We prove that a smooth toric Fano fibration admits a complete gradient shrinking Kähler-Ricci soliton.

We study the asymptotic behavior of Bergman kernels for high tensor powers of semipositive line bundles over Hermitian manifolds. At points where the curvature degenerates, the classical asymptotic expansion may fail. In this paper, we establish a full local asymptotic expansion and rapid off-diagonal decay near degenerate points at which the metric admits a local decoupled model. More generally, we make the following two spectral hypotheses: a localized mild spectral gap for the Kodaira Laplacian and a spectral gap for the rescaled local model. Under these assumptions, we prove a localization property and the rapid off-diagonal decay for the Bergman kernel. Furthermore, if the metric has a local quasi-homogeneous structure, we obtain a full local asymptotic expansion in the $C^\infty$-topology. As an application, we study pull-backs of positive line bundles under branched coverings. Near a smooth ramification hypersurface, the resulting asymptotic expansion reflects the branching order. Finally, for certain non-quasi-homogeneous models, we still obtain localization and leading-order asymptotics.

We establish an optimal leafwise width estimate for compact, spin, foliated bands with positive leafwise scalar curvature and infinite vertical $\widehat{A}$-cowaist. This result addresses a special case of the foliated version of Gromov's band width conjecture and generalizes the scalar and mean curvature comparison theorem for spin bands due to Cecchini and Zeidler. Our approach, which also applies to spin foliations, relies on a local boundary value problem for deformed sub-Dirac operators on the Connes fibration. A key technical ingredient is a Lichnerowicz-type formula in the adiabatic limit for almost isometric foliations, which captures the sharp coefficient $\operatorname{rank} F/(\operatorname{rank} F - 1)$ associated with the integrable subbundle $F$.

A Ševera bicomplex is a bicomplex $(M,d_{0},d_{1})$ whose differentials $d_{i}$ are Grothendieck differential operators of order $\leq i$ with respect to a given $A$-module structure on $M$, where $A$ is a graded commutative algebra. One also requires that $(M,d_{0})$ admits a contracting homotopy $K$ that is a differential operator of the same order as $d_{0}$. Under suitable assumptions, the differential $d_{2}$ of the second page of the spectral sequence associated with $(M,d_{0},d_{1})$ is a Batalin-Vilkovisky operator on $E_{2}$, so that when $E_{2}$ is a free rank 1 $A$-module, the graded commutative algebra structure of $A$ is enhanced to a Gerstenhaber algebra structure, and the choice of a $d_{2}$-closed basis element for $E_{2}$ further enhances this to a Batalin-Vilkovisky algebra structure. The prototypical example of this construction is Ševera's description of the Batalin–Vilkovisky algebra structure on the algebra of smooth functions of an odd symplectic manifold. Throughout the whole article we look at Batalin-Vilkovisky algebras through the lenses of Cartan calculus. Indications of a generalization to derived Cartan calculus are briefly discussed in the concluding section.

We construct free immersions of $P^3 \times S^1$ and $S^3 \times S^1$ into $14$-space. These are the first examples of free immersions in critical dimension of closed manifolds that are not spheres, projective spaces, tori, or surfaces. The idea is to mimic De Leo's construction for $m$-tori with $m \leq 5$. De Leo constructs a map that is symmetric with respect to a representation of the Lie group formed by the $(m - 1)$-torus; we replace this group with $\mathrm{SO}(3)$. In our examples, the osculating determinant is constant.

We prove that every smooth immersed disk in Euclidean three-space with nonzero constant mean curvature, regular up to the boundary and mapping its boundary diffeomorphically onto a circle, is an embedded spherical cap.

We prove the Brendle isoperimetric inequality for varifolds of finite mass and finite total first variation, allowing an arbitrary singular part. The distributional mean curvature is assumed only integrable. The result applies to integral varifolds and, more generally, to rectifiable varifolds with a positive lower density bound and weight measure concentrated on countably many $C^2$ submanifolds. The proof uses optimal transport and a locality theorem for the tangential Laplacian of an ambient convex function. The isoperimetric constant is sharp in codimensions one and two.

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+2(n-1)$ in the character variety depending on the Toledo value. Our construction uses $4$-cyclic Higgs bundles and their correspondence with $\partial$-alternating surfaces in complex pseudo-hyperbolic spaces. The associated equivariant minimal maps into the symmetric space are bi-Lipschitz embeddings. We further construct cocompact domains of discontinuity in the Shilov boundary. Their quotients are smooth fiber bundles over the surface with fiber homeomorphic to $S^{2n-1}\times S^{2n-1}$.

We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius $\pi/\sqrt{2}$ are $\mathcal{C}^{1,\frac{1}{2}}$-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius $\arctan(\sqrt{2})/\sqrt{2}$. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of $-1$ and is equivalent to the concavity of the domain.

Let $X$ be a compact Kähler manifold, let $D=\sum_iD_i$ be a simple normal crossing divisor, and fix $\beta_i\in(0,1)$. We introduce a parabolic structure $TX_*$ on $TX$, given along each $D_i$ by the filtration $0\subset TD_i\subset TX|_{D_i}$ with weight $w_i=1-\beta_i$ on the normal quotient. If $X\setminus D$ carries a Kähler–Einstein metric $\omega$ with cone angle $2\pi\beta_i$ along $D_i$, we show that $TX_*$ is parabolic polystable with respect to $[\omega]$. When the Einstein constant $\lambda$ is non-zero, we introduce parabolic structures on the Tian extension ($\lambda>0$) and on Simpson's canonical Higgs bundle ($\lambda<0$), and prove that these are polystable too. Combined with the Bogomolov–Gieseker inequality for parabolic bundles, this yields a Miyaoka–Yau inequality for these pairs.

In this paper we study conditions for the existence of a geometric quantization of a complete hyperKähler four manifold with an effective tri–Hamiltonian circle action and we show that the vanishing of the third coefficient $a_3$ associated to these metrics is equivalent to flatness.

For every closed oriented smooth four-manifold $M$ admitting a metric of strictly negative sectional curvature, we prove that $\dim_{\mathbb{Q}}H_1(T(M);\mathbb{Q})=\dim_{\mathbb{Q}}H_1(T_{\mathrm{TOP}}(M);\mathbb{Q})=\infty$ where $T(M)$ and $T_{\mathrm{TOP}}(M)$ denote the Torelli groups. This extends a theorem of Budney–Gabai for real hyperbolic manifolds.

Dirac-harmonic maps are a mathematical version of the supersymmetric non-linear sigma model of quantum field theory. Since they arise as critical points of an unbounded energy functional, it is a challenging task to establish general existence results. Within this manuscript we mainly prove a general existence result based on index theory for manifolds with boundary and apply it in various situations. Due to the presence of a boundary there are additional contributions in the Atiyah-Patodi-Singer index formula allowing for a non-vanishing index, which is the key argument in our proofs. Further explicit examples based on twistor spinors on surfaces are also presented.

We prove sharp two-eigenvalue isoperimetric inequalities for Neumann and Aharonov–Bohm Neumann spectra on surfaces. If $\Omega\subset\mathbb S^2$ is smooth, simply connected and proper, then the harmonic mean of $\mu_2(\Omega)$ and $\mu_3(\Omega)$ is bounded above by the first positive Neumann eigenvalue of the equal-area geodesic disk, with equality only for disks. For simply connected surfaces with Gaussian curvature bounded above, we obtain the magnetic analogue for the first two Aharonov–Bohm eigenvalues. The proof combines a two-dimensional reciprocal Rayleigh–Ritz principle with Green-level comparison. A key additional ingredient is a spectral ordering theorem for magnetic spherical caps: for $0<\nu<1/2$, the first two eigenvalues lie in the angular sectors of effective orders $\nu$ and $1-\nu$. We prove this by the factorization $L_0=T^*T$, $L_1=TT^*$ and an exact Neumann–Dirichlet spectral shift. We also obtain sharp full-sphere and closed-surface bounds, and an annular inequality in terms of conformal modulus and flux.

In this paper, we study the anisotropic-length-preserving weighted anisotropic curvature flow for smooth convex closed plane curves. For any smooth, embedded, closed, convex initial curve, the flow exists smoothly for all time, and the anisotropic curvature converges smoothly to that of the boundary of the corresponding homothetic Wulff shape as time tends to infinity.

In this paper, we establish an upper bound for the second eigenvalue of the scalar Jacobi operator of an arbitrary closed two-sided surface immersed in a contracted Berger sphere. The estimate involves the total squared mean curvature, the Euler characteristic, and an explicit nonpositive term determined by the angle between the surface normal and the Hopf direction. The proof combines the realization of a Berger sphere as a geodesic hypersurface of a complex projective plane, the first standard embedding of the latter into a Euclidean sphere, a weighted Hersch–Li–Yau balancing argument, and the conformal invariance of the Willmore functional. No minimality or constant mean curvature assumption is imposed. As an application, if $1/3\leq\alpha\leq1$ and the surface has nonpositive Euler characteristic, then its second Jacobi eigenvalue is nonpositive; it is strictly negative for $\alpha>1/3$. At the critical value $\alpha=1/3$, equality forces the immersed image to be congruent to the minimal Clifford torus; in the embedded category, this yields a complete characterization of the equality case.

We prove that a complete, two-sided, stable capillary minimal hypersurface in $\R^4_+$ is flat provided its contact angle $\theta$ satisfies $|\cot\theta|<\kappa_*$, where $\kappa_*=4\Gamma(3/4)^2/\Gamma(1/4)^2\approx 0.45$.

Motivated by results in Kähler geometry, in this work, we are interested in understanding the relation between integral curvature bounds and volume growth, under non-negative curvature in dimension three. In case of non-negative sectional curvature, we show that for metric on Euclidean space, it is of Euclidean volume growth if and only if it has average quadratic curvature decay. This is based on showing that metrics on three-dimensional Euclidean space with non-negative sectional curvature is of Euclidean volume growth if its asymptotic scaling invariant integral of scalar curvature is smaller than the sharp constant $8\pi$. We also show a gap Theorem if the curvature decay fast enough in the average sense, under non-negative Ricci curvature.

We introduce determinantal ellipticity, or det-ellipticity, a quantitative structural condition for fully nonlinear elliptic operators on compact Kähler manifolds that provides the link between the $L^\infty$-estimates and algebraic/combinatorial properties of a large class of Hessian elliptic operators. On the analytic side, we introduce $\mathcal D$-subsolutions extending the determinant sublevel condition of Sui–Sun. Using the auxiliary comparison method of Guo–Phong–Tong and pluripotential theory for complex Monge–Ampère equations, we obtain relative $L^\infty$-estimates in big cohomology classes under a determinant-entropy bound for viscosity supersolutions and singular reference potentials. Det-ellipticity provides a systematic construction of such subsolutions. On the algebraic side, we study Gårding elliptic polynomial operators of degree $d$. We characterize the determinant increment bound with exponent $d/n$ by positivity of the determinantal capacity of the top homogeneous part. This is equivalent to balanced-point conditions for the Newton polytopes of rank-one scalarizations and to slope semistability of an associated subspace polymatroid. Together with the analytic hypotheses, these criteria yield relative $L^\infty$ estimates for the corresponding fully nonlinear equations.

We prove that every stable weak critical point of the self-dual abelian Yang–Mills–Higgs energy on the round three-sphere is gauge equivalent to the vacuum, for every positive scale parameter. This establishes the three-dimensional statement proposed by Cheng. For the non-stable critical point, we give a quantitative upper bound for the lowest eigenvalue of the Hessian on the gauge quotient.

The focus of this paper is the curve shortening flow for closed spacelike curves in pseudo-Euclidean spaces, which has very few results so far. Will they produce singularities where certain tangent line tends to light cone? If not, will such a curve shrink to a circular point? To answer these questions, we establish a dichotomy for planar weighted normalized curve shortening flow with uniformly positive and bounded weights. Applying to closed smooth spacelike curves in pseudo-Euclidean spaces that admit a one-to-one convex projection onto a spacelike plane, at their finite maximal time we will see: either the curve shrinks to a point and becomes asymptotically circular, or the tangent directions subsequentially approach the null cone. Both alternatives occur. In the first case, this proves our previous conjecture that a strong spacelike curve in $\mathbb{R}^{2,q}$ with index 1 will converge to a circular point under the usual CSF. In the latter case, a monotone area-bivector defect is found in $\mathbb R^{2,1}$, which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.

We study the one-parameter family of Hessian metrics $g_\alpha=\alpha h_++(1-\alpha)h_-$, $\alpha\in[0,1]$, formed by convexly combining the log-partition functions (not the densities) of the multinomial and negative multinomial Fisher information Hessians. Four results are proved in full. (1) $g_\alpha\succ0$ for all $\alpha\in[0,1]$; its Gaussian curvature ($n=2$) is $-\tfrac14$ at $\alpha=0$ and $+\tfrac14$ at $\alpha=1$, but non-constant for intermediate $\alpha$, changing sign at a closed-form critical value $\alpha^\ast(w)$, with an analytic uniqueness proof. (2) Viewing $g_\alpha$ as the spatial metric of a Kasner-type volume equation, positive-definiteness forces one conserved quantity $\beta\le0$, while a second, independent quantity $C=4\det K$ governs the equation's branch (bounded oscillatory, parabolic, or unbounded growth) and is not fixed by $\beta$ or positive-definiteness alone; we show this discriminant dictionary is a theorem about the abstract matrix equation, to which $g_\alpha$ supplies initial data of either sign of $C$ without the trajectories remaining on $\{g_\alpha(\theta)\}$ itself. (3) $\Psi_\alpha$ is identified as the log-partition function of a compound distribution: a negative-binomial latent count $M$ (parameter $r=1-\alpha$), augmented by a Bernoulli parity bit and split multinomially, with non-negative base measure for every $\alpha$. (4) The associated Bregman divergence obeys a generalised Pythagorean theorem whose projection onto the symmetric locus is the arithmetic mean, independent of $\alpha$; the endpoints' curvature sign matches the Gauss–Bonnet excess of genuine geodesic triangles. Numerical checks are kept separate from analytic proofs throughout.