Papers

This is an updated version of a paper by the author which originally appeared in 1997. The original paper was a survey of the closely related fields of taut and Dupin submanifolds of Euclidean space, and this updated version includes many results in the field that have appeared since the publication of the original version. The emphasis is on stating results in their proper context and noting areas for future research, and relatively few proofs are given. The important class of isoparametric hypersurfaces is surveyed in detail, as is the relationship between the two concepts of taut and Dupin. Also included is a brief introduction to submanifold theory in Lie sphere geometry which is needed to state many known results on Dupin submanifolds accurately. The paper concludes with detailed descriptions of the main known classification results for both Dupin and taut submanifolds.

We prove that the critical value $2\pi^2$ is isolated for the Möbius cross energy of two-component links in the round three dimensional sphere $\mathbb{S}^{3}$. Specifically, there exists $\varepsilon_0>0$ such that any non-split, regular $H^2$ pair of curves with disjoint images, having vanishing first variation and energy at most $2\pi^2+\varepsilon_0$, must lie in the Möbius-reparametrization orbit of the standard Hopf link.

This paper investigates Yamabe-type soliton structures on the Sultana-Dyer (SD) black-hole spacetime. By employing the explicit metric and curvature quantities associated with the SD spacetime, we demonstrate that the non-Killing vector field $\partial_t$ induces an $h$-almost Yamabe soliton for explicitly determined functions $h$ and $\lambda$. We also establish the condition under which this structure reduces to an almost Yamabe soliton and examine why the coordinate vector fields $\partial_r$ and $\partial_\theta$ do not satisfy the corresponding Yamabe flow conditions. Furthermore, we show that the energy-momentum tensor, consisting of a timelike dust component and a null radiation fluid, is compatible with the Riemann, conformal (Weyl), concircular, conharmonic, and projective curvature structures. This compatibility reveals a strong geometric correspondence between the matter distribution and the causal structure of the spacetime. However, when the time-dependent scalar-curvature profile is subject to the constraints associated with late-stage cosmic expansion, the null, weak, and dominant energy conditions are violated. These findings provide a geometric characterization of the time-dependent Sultana-Dyer spacetime within the framework of generalized Yamabe solitons.

We prove that every closed $m$-quasi-Einstein manifold $(M^n,g,X,\lambda)$ with constant $\lambda\leq0$ is trivial whenever $m\leq-2$. This establishes the Colling–Dunajski conjecture in the range $m\leq-2$ and, for $n\geq5$, extends the previously known range $m\leq2-n$. This result is placed within a broader study of closed generalized $m$-quasi-Einstein manifolds $(M^n,g,X,\lambda)$. We establish differential and integral identities for such manifolds. These identities yield criteria for conformality, the Killing condition, and triviality, and provide a unified framework for several rigidity phenomena. After this, we recast the integrated Bochner formula as a Witten-type Hodge-energy identity. The cancellation of its quartic term at $m=-2$ leads to a triviality theorem in the generalized setting under a natural sign condition on the integral of $\langle X,\nabla\lambda\rangle$. An identity for the drift Laplacian gives a new proof of the previously known triviality result for $m\leq-n$ and extends it to generalized $m$-quasi-Einstein manifolds under a pointwise sign condition on $\langle X,\nabla\lambda\rangle$. Finally, complementing our criterion characterizing when a conformal potential field is Killing, we exhibit in the appendix a closed generalized $m$-quasi-Einstein manifold whose potential field is conformal but non-Killing.

Let $B\subset\mathbb{C}^n$ be a ball centred at the origin and $D=\{z^1=0\}$. Let $g$ be a Kähler metric of constant holomorphic sectional curvature (chsc) on $B\setminus D$, uniformly equivalent to the model cone metric of angle $2\pi\beta$, with $0<\beta<1$, and polyhomogeneous along $D$. We prove that, in suitable holomorphic coordinates $(w^1, \ldots, w^n)$ centred at the origin, the metric $g$ is the pullback of the corresponding complex space form by the map $(w^1,w^2,\dots,w^n)\longmapsto \bigl((w^1)^\beta,w^2,\dots,w^n\bigr)$.

The Landsberg Berwald conjecture asks whether every regular Landsberg metric is Berwald. We settle this conjecture for closed three-dimensional manifolds: every smooth strongly convex regular Landsberg metric on such a manifold is Berwald, without any reversibility assumption. Equivalently, there are no regular Landsberg "unicorns" on closed three-manifolds; this is a reformulation of the same conjecture, rather than a second independent conjecture.The proof combines a vanishing theorem for commuting Codazzi cubic tensors on closed surfaces, a rigidity theorem for three dimensional Minkowski norms with constant curvature indicatrices, and a rank one argument for the nonlinear curvature. The remaining R-quadratic case is settled by compactness along the geodesic flow. The fibrewise constant curvature theorem is an ingredient in this argument and does not assert the full Laugwitz conjecture.

For every $n\ge4$ and $L>0$, we construct a smooth $4$-PIC metric on $S^n$ with Urysohn $1$-width at least $L$ and an embedded stable minimal disk of intrinsic inradius at least $L$. These examples disprove the proposed width and stable-disk radius bounds under a positive lower bound for isotropic curvature. On closed even-dimensional manifolds, we prove the sharp estimate $\lambda_1^{(2)}\ge(n-1)\sigma/2$ under $\sigma$-PIC and show that equality forces roundness if a closed eigenform attaining the bound has rank at least four at some point.

We prove that every primitive harmonic $(n-1,1)$-form of a closed Kähler manifold of complex dimension $n$ vanishes provided its Kähler curvature operator is $\frac{n^2-n+2}{2}$-positive if $n \geq 4$, respectively $\frac{7}{2}$-positive if $n=3$. As a consequence, a closed Kähler manifold of complex dimension $n \geq 3$ with $3$-positive Kähler curvature operator is a real cohomology complex projective space.

We prove that for $n \geq 6$ there exists $\varepsilon (n)>0$ such that every closed orientable Riemannian manifold with $\left( \lceil \frac{n}{2} \rceil+ \varepsilon\right)$-positive curvature operator is a real homology sphere. For $n=6$ we show that the same result holds for manifolds with $4$-positive curvature operators.

We classify entire stable solutions with quadratic energy growth in the $4$-dimensional abelian Higgs model. More precisely, we prove that such solutions are holomorphic with respect to some complex structure, namely they satisfy the first order equations introduced by Bradlow, and consequently their nodal sets are holomorphic curves. Conversely, we construct solutions whose nodal set is any prescribed holomorphic curve with quadratic area growth. As a corollary, we show that all stable solutions with linear energy growth in $\mathbb{R}^3$ are two-dimensional Bogomolnyi vortices.

We study the spectral complexes arising from the de Rham complex on the Engel group $G$ viewed as a truncated multicomplex. We describe the two resulting complexes, analyse their compactly supported counterparts, and define the corresponding spectral currents by duality. We prove that the smooth spectral complexes embed naturally into the current complexes. Finally, we show that the Pansu pullback of a $W_{\mathrm{loc}}^{1,q}$ map $\varphi\colon G\to G$ with $q>7$ induces a morphism into the corresponding complexes of spectral currents.

Triple Aeppli–Bott–Chern–Massey products, shortly, triple ABC-Massey products are higher-order operations on the Bott–Chern and Aeppli cohomologies of a compact complex manifold, and their non-vanishing is an invariant of its pluripotential homotopy type. We prove that every non-Abelian complex unimodular solvable Lie algebra admits a non-vanishing triple ABC–Massey product, and we deduce that so does every compact complex parallelisable solvmanifold.

We construct a compact, smooth, simply connected complex threefold. Its second and third integral homology groups are cyclic of order two, and its remaining homology groups agree with those of the six-sphere.

We construct an explicit deformation of the bi-invariant metric on Sp(2) whose induced metrics on the Gromoll-Meyer exotic 7-sphere have positive sectional curvature for all sufficiently small positive parameters. The deformation is determined by a fixed self-adjoint operator compatible with the quotient action. The proof combines a uniform curvature estimate with an algebraic analysis of the horizontal commuting planes. On the critical set where the lower-order positive terms vanish, we establish an explicit positive lower bound for the cubic coefficient of the curvature numerator. Uniform control of the fourth-order remainder and a compactness argument then show that every sectional curvature is positive

For each integer $k\geq 1$, we study the $(k+1)$-dimensional helicoid $H_k\subset\mathbb{R}^{2k+1}$ parametrized by \[ (u_1,\ldots,u_k,s) \longmapsto \bigl(u_1e^{is},\ldots,u_ke^{is},s\bigr) \in \mathbb{C}^k\times\mathbb{R}\cong \mathbb{R}^{2k+1}. \] These helicoids form a basic and distinguished family of complete, properly embedded minimal submanifolds diffeomorphic to $\mathbb{R}^{k+1}$, and provide natural higher-dimensional analogues of the classical helicoid in $\mathbb{R}^3$. We completely determine their stability: $H_k$ is stable for $k\geq 3$ and unstable for $k\leq 2$. The sharp transition at $k=3$ is particularly striking: while the classical helicoid $(k=1)$ and its first higher-dimensional analogue $(k=2)$ are unstable, the four-dimensional helicoid $H_3\subset\mathbb{R}^7$ is already stable. For $k\geq 3$, we also determine their area-minimizing property: $H_k$ is area-minimizing when $k$ is even and not area-minimizing when $k$ is odd. The area-minimizing result is proved by constructing an explicit calibration, while the non-area-minimizing result follows from an explicit competitor. In particular, for every even $k\geq 4$, the $(k+1)$-dimensional helicoid $H_k$ is an entire minimal graph in $\mathbb{R}^{2k+1}$ that is area-minimizing.

We establish a dichotomy for Z/2-harmonic one-forms on Seifert fibered rational homology three-spheres. For an oriented base with at most three exceptional fibers, we prove nonexistence along any fixed connection-metric family when the fibers are sufficiently collapsed. With at least four exceptional fibers, existing results give existence for every Riemannian metric. We also complete the corresponding classification for nonorientable bases.

We establish sharp Sobolev-type geometric inequalities on $\mathbb{S}^n$ involving the total $\sigma_k$-curvatures $\int_{\mathbb{S}^n}\sigma_k(g)\,dv_g$. These results extend the optimal inequalities of Guan–Wang~\cite{GWDuke} from the cone $\mathcal{C}_k$ to the strictly larger cone $\mathcal{C}_{k-1}$, thereby enlarging the range of admissible conformal metrics. Our approach is variational and is implemented through a fully nonlinear conformal flow. Working in $\mathcal{C}_{k-1}$ introduces substantial analytic difficulties; in particular, one must obtain $C^2$ a priori estimates while simultaneously verifying that the flow remains parabolic. We resolve these issues via a carefully designed test function and by applying the maximum principle to the maximal eigenvalue of the Hessian matrix. As applications, we solve two open problems in dimensions 3 and 4. Finally, we give examples to show that these inequalities cannot be extended to $\mathcal{C}_{k-2}$.

Strict negativity of holomorphic bisectional curvature need not pass from the base and fibers of a compact holomorphic fibration to its total space, even when the Kodaira-Spencer map is everywhere injective. We construct a compact relative Kähler fibration over a genus-two curve, with a smooth projective threefold as total space and an everywhere injective Kodaira–Spencer map, whose induced fiber metrics have strictly negative holomorphic bisectional curvature, whereas the total space admits no Kähler metric with this curvature property. This gives a negative answer to an open problem posed by To and Yeung. We also construct compact, effectively parametrized Monge-Ampère fibrations from universal quaternionic abelian surfaces over Shimura curves. For products of these families, the generalized Weil-Petersson metric has nonpositive holomorphic bisectional curvature and strictly negative holomorphic sectional curvature, but its mixed bisectional curvatures vanish. Thus strict bisectional negativity can fail both for the existence of a metric on the total space and for the natural metric on the parameter space, despite effective variation at every point.

We study complete static manifolds with boundary admitting a nowhere-vanishing static potential. Our main result shows that, under a natural lower bound relating the scalar curvature and the boundary mean curvature, a simple static manifold with boundary must in fact have positive scalar curvature, negative boundary mean curvature, and be compact; we also obtain explicit relations and estimates involving the volume of the manifold and the geometry of its boundary. In the scalar-flat case, we prove global splitting and Ricci-flat rigidity results, including for disconnected boundary, while in the negative scalar curvature case we establish a sharp mean-curvature bound and characterize the equality case by an exponential warped-product structure. The proofs rely essentially on the study of the associated Einstein manifold. In appendix we derive several identities for static manifolds with boundary and discuss the associated Einstein manifold technique in the boundaryless setting.

We ask how much of the geometry a charged particle moves in is fixed by its motion, and how much a particle must bring. We ask of a symplectic structure only that it relate velocity to momentum as Hamilton's equations do, and we ask it of every energy at once. In particular, we show that the structures meeting that demand are the canonical one and its twists by a closed two-form of the base. A field provides the two-form, a particle the multiplier before it, which we identify constitutively with its charge. Thus, a single energy governs a family of structures, and each particle takes the one its charge fixes. We then ask what a particle must bring to be given a momentum, and we show that the degree of that map settles the degree at which a field enters Newton's Second Law. An antisymmetric bilinear form returns no Lorentz force, whilst a Randers metric returns one — a length whose difference from a Riemannian one is linear in the velocity. Moreover, we find that metric already within the twisted structure, as its primitive over a level of the free energy, and its law of transport to be nonlinear, no affine connection being known to serve. Under an indefinite signature the length parts from the dynamics, and the extremals turn from shortest to longest. On the round sphere a monopole flux leaves no such metric, whilst the transport remains and prequantisation, given a unit of action, restricts the charge to a lattice. In this manner, we conclude that each charge-to-mass ratio receives a geometry of its own, so that by a functionalist criterion none of them is the spacetime of a charged particle.

We prove the black hole uniqueness conjecture in the axially symmetric, stationary, electrovacuum setting, subject to the refined asymptotic analysis of the associated singular harmonic maps, which includes an analyticity hypothesis at the axes. More precisely, it is shown that any asymptotically flat solution of the Einstein–Maxwell equations in this class, with more than one black hole horizon component is either: Majumdar–Papapetrou, up to a duality rotation, in which case all logarithmic angle defects vanish, or every finite axis rod logarithmic angle defect is strictly negative and hence every interaction force is strictly attractive. The proof extends the singular harmonic map method used for vacuum Kerr uniqueness in [18].

We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved via integrals over pairs of boundary points, in the spirit of Banchoff-Pohl, together with an estimate for Jacobi fields along geodesic chords. The inequality persists for boundaries of isoperimetric regions in geodesic balls, whose mean curvature is constant only on the free part. An isoperimetric-profile argument, after Kleiner, completes the proof. Our method also gives a new proof in dimension $3$.

We introduce and study a scalar $q$-deformation of the cross-ratio on $\mathbb P^1(\mathbb Q)$. Our construction is based on the notion of $q$-deformed rational numbers due to Morier-Genoud and the author. The $q$-cross-ratio is invariant under $\mathrm{PSL}(2,\mathbb{Z})$, while elements of determinant $-1$ of $\mathrm{PGL}(2,\mathbb{Z})$ act by $q\mapsto q^{-1}$. A principal result is its relation to $q$-deformed Coxeter friezes associated with rational polygons. The expansion at $q=e^h$ yields an algebraically independent sequence of modular invariants and relative invariants, although this sequence does not separate modular orbits. We compute the first two nonconstant coefficients of this expansion explicitly.

We study profinite rigidity of real hyperbolic lattices, with the main application in dimension three. The basic tool is a profinite version of the dual cube complex of a pair of filling curves on a surface, which we call a profinite cubical residue complex. We prove that an equivariant isomorphism between the completions of two such complexes is induced, after conjugation by a single element, by an isomorphism of the discrete groups and complexes, and that the same holds for free cocompact cellular actions of finitely generated residually finite groups on connected, locally finite, finite dimensional regular CW complexes. On a closed orientable surface of genus at least two, a theory of divisible fillings determines the completed cut tree of a nonseparating simple closed curve. Combined with the homological intersection criterion of Boggi and Zalesskii, this determines the completed dual square complex of a nonseparating filling pair, and hence realizes every isomorphism of profinite surface groups which preserves the two marked cyclic subgroups. In dimension three we combine these results with virtual fibering, Liu's profinite invariants of hyperbolic $3$-manifolds and the goodness of $3$-manifold groups. Massey products and virtual domination give cohomological integrality, and pseudo-Anosov dynamics, periodic orbit traces and cross sections of suspension flows give a correspondence of prime periodic orbits together with the fiber curves needed for the realization. Profinite rigidity of closed and of cusped lattices follows, the cusped case through cyclic orbifold fillings, and we obtain $\Out(\Gamma)\cong\Out(\wh\Gamma)$ for every lattice of $\PSL_2(\C)$.

Let $\Lambda$ be a compact Bohr–Sommerfeld Lagrangian submanifold of a compact Kähler manifold equipped with a holomorphic prequantum line bundle. We study the asymptotic expansion of the Lagrangian states associated with $\Lambda$. In particular, we compute explicitly the first nontrivial correction term and show that it is expressed in terms of geometric invariants of the ambient Kähler manifold and the Lagrangian submanifold, including their scalar curvatures, the second fundamental form, and the mean curvature. As a consequence, we obtain the corresponding second-order asymptotic formula for the $L^2$-norm of the Lagrangian states.

The twistor correspondence provides a duality between curves in $S^4$ and ruled surfaces in $\mathbb{CP}^3$ in which the conformal properties of the former are reflected in the projective properties of the latter. We use this to characterise a $14$–dimensional family of homogeneous ruled surfaces in $\mathbb{CP}^3$ which correspond to loxodromes in $S^4$.

We resolve Case's question (Crelle's Journal, 2024) on the sharp lower threshold for Obata-type rigidity of $I_a=Q+a\sigma_2(A)$ in positive Einstein conformal classes of dimension $n\ge4$. Using a new reference-curvature identity, we prove that, on a closed connected manifold, every smooth metric in such a class with nonnegative scalar curvature and constant $I_a$ is Einstein whenever $a\ge-4$. The scalar-curvature sign assumption can be removed for $-4\le a\le-2(n-2)/(n-1)$. We also extend the quotient rigidity theorem of Ge–Wang–Wei to $I_a=\Lambda R^\theta$ for $R>0$, $a\ge-4$, and $\theta\le1$. Sharpness is established in every dimension by a shooting construction. For each sufficiently small $\eta>0$, we construct a smooth non-Einstein metric in the round conformal class on $\mathbb S^n$ with positive scalar curvature and the same constant $I_{-4-\eta}$ as the unit round metric. The construction smoothly matches a perturbed Schwarzschild neck near the equator with perturbed round caps at both poles.

Using a reference-curvature technique, we prove four main theorems. (i) For $n\geq4$, we remove the mean-curvature pinching from the Case-Wang rigidity theorem for $\sigma_2$-flat conformal hemispheres with $A_g\in\overline{\Gamma_2^+}$ and positive constant natural boundary curvature. (ii) On every closed positive Einstein manifold of dimension $n\geq5$, each conformal metric of constant $\sigma_2$-curvature is Einstein, and is homothetic to the background unless the background is a round sphere, thereby resolving the $\sigma_2$-Obata problem posed by Chang, Gursky, and Yang. (iii) We extend the spherical constant-$\sigma_2/\sigma_1$ classification and sharp integral comparison to closed positive Einstein manifolds. The comparison holds throughout the positive-scalar-curvature conformal class, with equality only for homotheties on nonround backgrounds. (iv) We establish global quantitative $\sigma_2$ stability on every fixed nonround positive Einstein background of dimension $n\geq5$, throughout its positive-scalar-curvature conformal class.

Let $X$ be a connected oriented even-dimensional manifold whose universal cover is spin. We prove that $X$ admits no complete metric with uniformly positive scalar curvature when either infinite relative $K$-area or a relative cohomological condition holds along compact sets escaping to infinity in a fixed open subset with compact complement. The proof compares twisted Dirac operators on spin covers of compact manifolds with boundary and combines a heat-kernel argument on fundamental domains with a long-neck estimate.

We give counterexamples to Yau's question on normalized scalar curvature integrals in every dimension $n\geq4$. For each such $n$, there exists a smooth complete Riemannian metric $g$ on $\mathbb{R}^n$ with nonnegative Ricci curvature and a pole such that \[ \lim_{R\to\infty}R^{2-n}\int_{B_q(R)}\mathrm{Scal}_g\,\mathrm dV_g=+\infty \] for every fixed $q\in\mathbb{R}^n$. Here $B_q(R)$ denotes the geodesic ball of radius $R$ centered at $q$, and $\mathrm{Scal}_g$ is the scalar curvature of $g$.