Papers

We study global and variational properties of $\mathbb R$-equivariant minimal Lagrangian surfaces in the complex quadric $Q_2$ and the complex hyperbolic quadric $Q_2^*$ arising from loop group potentials. In the $Q_2^*$ case, we work under the timelike-axis condition throughout the associated $\mathbb{S}^1$-family. Excluding the flat case in $Q_2$, we give a necessary and sufficient criterion for closing in the $y$-direction: a member $f^{\lambda_0}$ closes if and only if the spectral ratio $\mu_2/\mu_1$ is a positive rational number. The closing parameters form a dense countable subset of $\mathbb{S}^1$, while the non-closing parameters are dense and have full arc-length measure. The closing members yield cylindrical quotients in $Q_2$ and annular quotients in $Q_2^*$. Apart from the flat case in $Q_2$, the surfaces under consideration have infinite absolute total curvature. The equivariant surfaces considered here in $Q_2$ are unstable, whereas minimal Lagrangian surfaces in $Q_2^*$ are stable. Finally, for compact cylindrical domains of the $y$-closing surfaces in $Q_2$, we obtain explicit sufficient width conditions for Hamiltonian stability and instability under variations fixed near the boundary. We also establish the existence of a finite geometric critical width and determine it explicitly in the flat case.

In our previous papers [33, 34], we established the affine Bernstein theorem and proved the interior regularity for the affine Plateau problem in two dimensions. In this paper, we prove the interior regularity for the affine Plateau problem in all dimensions and present a smooth counterexample to the affine Bernstein problem in dimensions ten and higher. We also prove an affine Bernstein theorem in all dimensions under a growth condition.

The centroaffine Bernstein conjecture for hyperbolic centroaffine extremal hypersurfaces predicts that nonnegative Ricci curvature together with either Euclidean or centroaffine completeness forces the hypersurface to lie in Wang's class. We disprove it by systematically studying Calabi compositions: we develop a general geometric method for constructing complete centroaffine extremal hypersurfaces and apply it to construct two families that violate this rigidity. The first family is Euclidean complete but has incomplete centroaffine metric, nonnegative Ricci curvature, and positive Gauss curvature in dimension two; iterating the composition with a point gives examples in every dimension $n\ge2$. The second family is flat and complete in both metrics in dimension three, and iteration yields examples in every dimension $n\ge3$. Thus the centroaffine Bernstein conjecture fails under Euclidean completeness for $n\ge2$ and under centroaffine completeness for $n\ge3$.

We study the fundamental groups of a normal projective Calabi-Yau variety $X$ with klt singularities and of its regular locus $X_{reg}$. We prove that both groups are almost abelian, and that their ranks are determined by the irregularities of finite etale and quasi-etale covers, respectively. Our approach is differential geometric, and relies on recent advances in the geometric theory of complex Monge-Ampere equations.

In one complex variable the Schwarzian derivative characterizes the Möbius transformations, arises from a second-order linear equation, and controls univalence through the theorems of Nehari and of Ahlfors–Weill. This chapter presents what is known about these three properties for locally biholomorphic mappings in $\mathbb{C}^n$. Starting from Oda's definition we construct a bilinear operator whose norm, taken with respect to the Bergman metric, is invariant under the automorphisms of the ball, and we show that this operator measures the extrinsic curvature of the preimages of hyperplanes. We study the linearly invariant family defined by a bound on this norm, we prove that a sufficiently small bound implies univalence, and we describe the extension of Ahlfors–Weill type, which takes values in the space of hyperplanes. The polydisk and convex mappings are also treated, and open problems and exercises are included.

We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus $\mathbb{T}^n\hookrightarrow\mathbb{R}^{n+1}$. Due to the recursive structure we build an exact uniform sampler by generalizing the process from the curved 2-torus. We generalize projected normal distributions to the curved $n$-torus and we develop inference theory for these distributions. We utilize both maximum-log-likelihood estimates and energy-based Fréchet estimates for the projected normal distribution for this purpose. This is done by establishing identifiability for isotropic projected normal distributions and by applying theory of maximum likelihood and intrinsic statistics for the curved tori. We numerically verify the geometry, sampler, density transformation, and energy solver. Lastly, we demonstrate the applicability of the proposed model using real data.

We show that Hitchin-Wolf's harmonic map parametrization of Teichmüller space, the Schwarzian holonomy map, and Uhlenbeck's almost-Fuchsian holonomy map all arise as restrictions of a single locally biholomorphic map into the $\mathrm{PSL}(2,\mathbb{C})$ character variety. The map admits a right inverse defined on an open subset of quasi-Fuchsian space, encoding harmonic maps together with the Bers embedding. The construction goes through complex harmonic maps to the space of oriented geodesics of $\mathbb{H}^3$. Along the way, we further develop the theory of complex harmonic maps, with a focus on the natural $\mathbb{C}^*$-action and the relation to complex projective structures. This leads to an independent application: we obtain a new geometric interpretation of certain real twistor lines in the rank-two Deligne-Hitchin moduli space and we recover a theorem of Gothen-Silva.

In this paper, we study joint deformations of pairs $(X,E)$, where $X$ is a compact complex manifold and $E\rightarrow X$ is a holomorphic vector bundle. We prove that $(X,E)$ has unobstructed deformations if $X$ is a Fujiki manifold with torsion canonical bundle and $H^2(X,\textrm{End}^0E)=0$, where $\textrm{End}^0 E$ denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak $\partial\bar\partial$-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at $E_1$. These results provide non-Kähler extensions of the theorems of Li–Pan and Iacono–Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi–Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.

We study the Yamabe–Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions $3\leq n\leq 14$. We show that this dimensional threshold $n_*=14$ is sharp by constructing in $n\geq 15$ nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, $n_*^{LCF}=18$, and the metrics with umbilic boundary, $n_*^{umb} =20$. In all three settings we also prove that the Weyl–umbilicity vanishing conjecture holds for $4\leq n\leq n_*$ and construct counterexamples in $n>n_*$ by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe–Escobar equation, outside the conformal class of the round hemisphere.

We show that if an open smooth $6$-manifold $M^6$ is the interior of a compact, contractible manifold with boundary $X^6$, such that $\pi_3(X,\partial X)=0$, and $M$ supports a complete metric with uniform positive BiRicci curvature and Ricci curvature bounded below, then $M$ is diffeomorphic to $\mathbb{R}^6$. Additionally, we show that if an open smooth $n$-manifold $M^n$, $n\in\{6,7\}$, is the interior of a compact, contractible manifold with boundary $X^n$, such that $\pi_3(X,\partial X)=0$, and $M$ supports a complete metric with positive BiRicci curvature with at most $C$-quadratic decay at infinity for some $C>C_n$ and nonnegative Ricci curvature, then $M$ is diffeomorphic to $\mathbb{R}^n$. Lastly, for each $n\geq6$, we construct a compact, contractible Newman $n$-manifold, $N^n$, based on a presentation of the Higman group such that $\pi_i(N,\partial N)=0$, $3\leq i \leq n-3$, and $\mathrm{int}(N)$ admits a complete metric with uniform positive scalar curvature and Ricci curvature bounded below.

Anisotropic Voronoi diagrams, in which each cell is the region first reached by an ellipse growing from its site, generate non-convex cells whose integral geometry can be described using Minkowski tensors of arbitrary order. In this work, we derive semi-analytical formulas for Minkowski tensors of arbitrary order of the 0th, 1st and 2nd kind for planar anisotropic Voronoi cells that are star convex at their nucleation point. The resulting formulas involve integrals of contact functions, which describe the cell boundary through a diffeomorphic transformation. This transformation renders the geometry amenable to analytical treatment, enabling us to obtain expressions for the desired Minkowski tensors. Although most of the integrals lack closed-form solutions, they can be efficiently approximated numerically once the contact function and its derivatives are known along the boundary of the cell of interest.

We study the critical points of a quadratic functional depending on the gradient of the Bach tensor on Riemannian four-manifolds, which generalize the Bach-flat condition. We show that, on every closed four-manifold, there exists a weak Bach-parallel metric, i.e. a critical point for this functional with respect to conformal variations: in particular, we prove that there exist infinitely many conformal classes which contain a unique minimizer for the functional, up to constant positive rescaling. Next, we analyze the global minima of the functional, i.e. metrics with parallel Bach tensor, relating these metrics to well-known variational problems. Using a version of de Rham's splitting theorem on complete four-manifolds, we provide a classification result for products of surfaces, exploiting the theory of conformal gradient solitons; we also construct a new explicit example of a Bach-flat metric which is neither locally conformally flat nor conformally Einstein and we characterize HCMU metrics on complete surfaces. Finally, we prove an equivalence between the Bach-parallel condition on 4D cylinders and the existence of critical metrics for a well-known quadratic curvature functional in dimension three: in this direction, we also prove a characterization of flat three-manifolds, under some curvature and finite energy assumptions.

Extending the well-known Riemannian positive mass theorem to low-regularity metrics is an ongoing effort, starting from the early 2000s, that has recently seen renewed interest. Contrary to many other approaches we do not wish to assume smoothness at infinity nor to introduce a new definition of mass tailored to our approach, but rather go back to the coordinate isoperimetric mass originally proposed by Huisken. We establish new asymptotic expressions for the coordinate isoperimetric mass of continuous asymptotically flat metrics, which constitute the main ingredient in proving a finiteness criterion, coordinate invariance and equality to the generalized ADM mass of Lee and LeFloch under reasonable assumptions. In particular, for $W^{1,p}_{-\tau}$ asymptotically flat metrics, with $p>n$, $\tau>(n-2)/2$, and nonnegative distributional scalar curvature, the coordinate isoperimetric mass is indeed coordinate invariant, and we establish a positive mass theorem in this regularity without any topological assumptions aside from asymptotic flatness. Our proof relies on a density theorem similar to that of Lee, Lesourd, and Unger, but under a weaker regularity and decay assumption, which may be of independent interest.

Let $M$ be a closed connected Riemannian manifold and let $\mathcal P_s$ be the fractional perimeter on $M\times\mathbb R^k$ defined through its heat semigroup, where $k\ge1$ and $0<s<1$. We prove that the straight cylinder $M\times B_R$, with $B_R\subset\mathbb R^k$ a ball of radius $R$, uniquely minimizes $\mathcal P_s$ among all measurable sets of the same volume, up to translations in $\mathbb R^k$ and null sets, whenever $R\sqrt{\lambda_1(M)}$ is at least an explicit threshold depending only on $k$ and $s$. Here $\lambda_1(M)$ is the first positive eigenvalue of the nonnegative Laplace–Beltrami operator $-\Delta_M$ on $M$. Under a larger threshold, the perimeter deficit controls both the deviation of the slice volumes from their mean and the square of the asymmetry relative to straight cylinders. For flat tori this gives large-volume uniqueness of straight cylinders for the periodic fractional perimeter, including singly periodic sets in every ambient dimension. The proof combines spectral reduction on $M$ with rearrangement of the Euclidean slices and a one-variable comparison between an interaction gain and a concavity loss.

We introduce a continuous-time counterpart of the median filter that gradually denoises a given image. In the limit of vanishing stencil size, our results show that the evolution converges to level-set mean curvature flow. Surprisingly, and for the first time for any median-type filter scheme, we can prove the convergence of energies in this limit. Such strong convergence results are of major interest as they often have to be assumed in the literature to prove convergence of related schemes. Moreover, this result carries crucial geometric information, namely the unit multiplicity of interfaces. The proof relies on a compensated compactness argument inspired by the work of Evans and Spruck.

We construct an example of a smooth embedding of $\mathbb{S}^{p+q-1}$ in $\mathbb{R}^{p+q}$ whose evolution under the mean curvature flow forms an isolated singularity at one point, after which it is a smoothly embedded copy of $\mathbb{S}^{p-1}\times \mathbb{S}^q$. In doing so, we prove some general results about $SO(p)\times SO(q)$-invariant mean curvature flow whose initial data is generated by the rotation of a graphical profile curve, including an analysis of bubblesheet singularities. The most involved part of our analysis is an instant smoothness result, which requires a delicate approximation and pseudolocality.

Let $H>0$ and $\kappa,\tau\in\mathbb{R}$ satisfy $4H^2+\kappa>0$ and $\kappa\leq 4\tau^2$. We prove that an immersed stable disk of constant mean curvature $H$ and circular boundary in the space $\mathbb{E}(\kappa,\tau)$ is a spherical cap of the canonical rotational $H$-sphere.

In this paper, we study strong uniqueness for smooth instantaneously complete Chern-Ricci flows on noncompact complex manifolds. Under suitable reference metric and positivity assumptions, we establish strong uniqueness for initial Hermitian metrics which are Kähler outside a compact set. The initial metric may be incomplete, and no curvature, metric comparison, or growth bounds are imposed on the evolving solutions. In particular, we obtain strong uniqueness from complete initial metrics in this class with bounded Chern-Ricci curvature. We also prove stationarity of flows from the Euclidean metric on $\mathbb{C}^n$ and uniqueness of the instantaneously complete flow from the Euclidean metric on the complex unit ball for all positive time. Finally, on complex surfaces, we show that a Ricci flow remaining Hermitian for a fixed complex structure is Kähler at all times if it is Kähler at one time, and derive a corresponding strong uniqueness result for Ricci flows in this class.

The mean exit time of a Brownian motion from a tube around a closed, minimal embedded \(n\)-dimen\-sional submanifold of the \((n+m)\)-dimen\-sional sphere is studied, establishing two-sided bounds via radial transplant. In accordance with known results when \(n = 0\) or \(m = 1,\) the mean exit time goes to infinity as \(n \to \infty\) and to \(0\) as \(m \to \infty.\) Simultaneously, bounds for the volume of such tubes are derived and compared to Weyl's tube formula.

In this paper, we prove a uniform approximation theorem with interpolation for complete non-orientable minimal surfaces of finite total curvature in the Euclidean space $\mathbb{R}^n$ $(n\ge 3)$. In particular, we establish the existence of such surfaces with the complex structure of any open conformal non-orientable surface of finite topology and parabolic type.

We prove that if a compact Kähler manifold admits a Hermitian metric with quasi-negative holomorphic sectional curvature for the Chern connection, then its canonical bundle is ample. The Hermitian metric realizing this curvature need not be Kähler. This proves the Kähler case of the quasi-negative conjecture of Yang and Zheng.

In this paper we investigate biconservative and biharmonic hypersurfaces in product spaces $L^m(\varepsilon)\times {\mathbb R}$, where $L^m(\varepsilon)$ denotes the space form ${\mathbb S}^m$ or ${\mathbb H}^m$ of constant sectional curvature $\varepsilon=1$ or $\varepsilon=-1$ respectively. We obtain their complete description in the case of isoparametric hypersurfaces and hypersurfaces of constant angle. Moreover, we prove some classification results for biconservative and biharmonic hypersurfaces for which the gradient of the mean curvature function is a principal direction. In this context we obtain new examples of complete, non-cylindrical, non-totally umbilical biconservative hypersurfaces in ${\mathbb H}^m\times {\mathbb R}$. Then we prove a rigidity theorem related to these examples. Finally, we prove a global non-existence theorem for complete proper biharmonic hypersurfaces in ${\mathbb H}^m \times {\mathbb R}$ such that the square of the mean curvature function is bounded and the Ricci curvature is bounded from below.

This work presents a prototype implementation of a noncoherent multiple-input multiple-output (MIMO) system based on Grassmannian signalling, where neither the transmitter nor receiver requires channel state information (CSI), operating at a carrier frequency of 15 GHz in the frequency range 3 (FR3) upper mid-band envisioned for 6G. To the best of our knowledge, this is the first prototype implementation of Grassmannian signaling reported in the literature beyond sub-6 GHz bands. We identify the motivation of applying Grassmannian signaling, a form of unitary space-time modulation (USTM), in this band for wireless and mobile applications with low spectral-efficiency requirements but strict reliability and power-efficiency demands. With this idea in mind, we present a system design integrated with orthogonal frequency-division multiplexing (OFDM) that addresses the challenges associated with the higher carrier frequencies of FR3 while providing reliable operation across a broad range of scenarios, including urban and rural environments as well as vehicular communications. We then present the prototype implementation of the proposed design, based on software-defined radio (SDR) devices integrated with radio-frequency (RF) mixers, filters, and power amplifiers to ensure adequate operation in the targeted band. Over-the-air (OTA) experiments based on symbol error rate (SER) measurements under non-line-of-sight (NLoS) and line-of-sight (LoS) scenarios, and across different data rates, show good agreement with software simulations, thereby validating the prototype implementation.

We construct smooth Einstein metrics with positive scalar curvature on $\#_k({\mathbb S^2}\times {\mathbb S^2})$ and on $k\mathbb{CP}^2\#(2k+1)\overline{\mathbb{CP}}{}^{2}$, for $k\geqslant2$. In particular, we show that positive Einstein metrics exist on infinitely many homeomorphism types of closed $4$-manifolds. The first family shows that any simply connected spin $4$-manifold carrying metrics with positive scalar curvature is homeomorphic to an Einstein $4$-manifold. Wick rotation gives associated Lorentzian static axisymmetric Einstein metrics with positive cosmological constant and arbitrarily many horizons with the same surface gravity. The metrics are obtained by varying the cone angle $2\pi\beta$ of a prescribed singularity along a torus, starting from the product metric on $\mathbb S^2\times\mathbb S^2$ and from the symmetric Kähler-Einstein metric on $\mathbb{CP}^2\#3\overline{\mathbb{CP}}{}^2$. At $\beta=\frac{2}{k+1}$, changing the period of the angle around the torus removes the cone singularity and changes the topology. For both families, as $k\to+\infty$, the metrics collapse along the torus orbits to an explicit $2$-dimensional disk, whose boundary has Hausdorff dimension $\frac{4}{3}$. At the scale of curvature, they converge to complete periodic Ricci-flat metrics with Kasner asymptotics: the Euclidean Myers-Korotkin-Nicolai metric for the first family, and a new one for the second.

We construct a smooth embedded open annulus in Euclidean three-space with strictly negative Gaussian curvature and injective Gauss map that contains a simple closed asymptotic curve of zero normal self-linking. This gives an affirmative answer to Problem 1.1 of Ghomi and Raffaelli. The construction starts with two simple planar curves whose tangent vectors are positive multiples of one another. Their mixed integral is made zero by rounding a pair of polygons with one free length parameter.

For every integer $N\ge4$, we construct a fixed smooth, non-locally-conformally-flat metric on the $n$-dimensional unit sphere $\mathbb{S}^n$ for which the constant $Q_{2N}$-curvature equation admits an $L^\infty$-unbounded sequence of positive solutions. The construction applies for $n\ge2N+20$ when $N=4,5$, $n\ge2N+19$ when $6\le N\le8$, $n\ge2N+18$ when $9\le N\le17$, and $n\ge2N+17$ when $N\ge18$. For each $L \in \mathbb{N} \cup \{0\}$, the metric can be chosen arbitrarily close to the round metric in the $C^L$ norm. Together with the known cases $N=1,2,3$, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total $Q_{2N}$-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying $\operatorname{Ric}=(n-1)g$. Building on Juhl's formulas, we identify this Hessian, for every $N\in\mathbb{N}$ and $n>2N$, as a degree-$N$ polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real $n$, this matrix changes from negative definite to indefinite at $n_N=2N+a_*+c_*N^{-1}+O(N^{-2})$ as $N\to\infty$, with $a_*\approx16.201871$ and $c_*\approx13.512880$, explaining the eventual bound $n\ge2N+17$.

We prove that finite-dimensional geodesic locally compact Alexandrov spaces with curvature bounded below (CBB), and in particular complete Riemannian manifolds with lower bounded sectional curvature, satisfy a Besicovitch covering theorem by showing that they are directionally limited in the sense of Federer.

We study the HKT geometry of compact locally conformally hyperkähler manifolds. We prove that on these manifolds the quaternionic Monge-Ampère equation is always solvable, extending results of Alesker and of Dinew-Sroka, and we confirm the conjecture on the maximal existence time of the HKT-Ricci flow. On quaternionic Hopf surfaces we construct non-trivial HKT solitons and prove convergence of the normalized flow for symmetric initial data.

In current work, we delve into the study of complex Hessian equations on compact Kähler manifolds. By imposing a constraint on both the bisectional curvature and the oscillation of the solution, we demonstrate that the Laplacian estimate can be linearly controlled by its gradient estimate. Consequently, we are able to derive gradient estimate for the solution using a standard interpolation argument, without the need to employ the blow-up technique. As an application, we demonstrate that when the right-hand side of the complex Hessian equations approximate 1 in the $L^1$ norm, the desired gradient estimate is obtained.

For a proper cocompact action of a finitely generated discrete group $\Gamma$ on a spin manifold $M$, the higher localised $\widehat{A}$-genera $\widehat{A}_g(M,\omega)$ introduced earlier by the author and Mathai were shown to be an obstruction to $\Gamma$-invariant metrics of positive scalar curvature, for $\omega$ arising from the subring of $H^*(\underline{B}\Gamma,\mathbb{R})$ generated by elements of degree at most $2$, under the assumptions that the fixed-point set $M^g$ is connected and that $g$ is regular with respect to a certain real group $2$-cocycle. We show that both assumptions can be dropped.