Homogeneous spaces

In this paper, we first prove Appleton's conjecuture on the uniqueness of noncollapsed steady solitons constructed in. We also classify non-Ricci-flat $\kappa$-noncollapsed cohomogeneity one steady gradient Ricci solitons in dimension four and five. After passing to a finite cover, in dimension four, they must be the Bryant soliton or Appleton's noncollapsed 4d steady soliton; in dimension five, they must be the Bryant soliton or solitons constructed by Ivey–Dancer–Wang.

We study curvature flows for the dual Orlicz–Minkowski problem for strictly convex capillary hypersurfaces in the Euclidean half-space. The prescribed function has the form $\phi (h)G(X)$, where $X=\nabla h+h\nu$ and $G$ may be nonradial and nonhomogeneous. For acute contact angles, the evenness assumption together with either of the two integral conditions yields a smooth weighted-volume-preserving flow for all $t>0$ and a smooth solution to the dual Orlicz–Minkowski equation. Without the evenness assumption, two strict spherical-cap barriers yield the same conclusions for an unnormalized flow, whose limiting hypersurfaces solve the dual Orlicz–Minkowski equation with $c=1$. A uniform strict monotonicity condition under rescaling ensures uniqueness for $c=1$ and convergence of the unnormalized flow in any $C^k$ norm at an exponential rate. The problem considered here includes the capillary $L_p$-Minkowski and $L_p$ dual Minkowski problems as special cases.

We show that Ricci flows of invariant metrics on twisted principal bundles with nilpotent fibres over $S^1$ are immortal and type III. After normalisation and lift to the universal cover, a sequence of metrics converges in the Cheeger–Gromov topology to a homogeneous expanding Ricci soliton. The proof extends the bracket flow technique from homogeneous Ricci flows to invariant Ricci flows on twisted principal bundles.

We prove that on a connected, simply-connected nilpotent Lie group, the notions of left-invariant generalized Ricci soliton, semi-algebraic and algebraic generalized Ricci soliton coincide. Moreover, every such soliton has harmonic torsion. We classify all algebraic solitons with positive definite metric on the five-dimensional Heisenberg algebra, the four-dimensional oscillator algebra and the four-dimensional split-oscillator algebra up to automorphisms and rescalings. We find that there is only a finite number of algebraic solitons, all of which have harmonic torsion.

We classify complete, non-compact, simply connected, irreducible four-dimensional steady gradient Ricci solitons that are globally conformally Kähler and have an isoparametric potential. The steady Ricci solitons are $U(2)$-invariant nuts on ${\mathbb{R}}^4$, including the Bryant and Cao solitons, and bolts on ${\mathbb{S}}^2\times {\mathbb{R}}^2$ and on $\mathcal{O}(-m)\to {\mathbb{CP}}^1$. For each $m\ge 3$ there is exactly one non-collapsed bolt. A key step, valid for Ricci solitons of every sign, is that away from the locally conformally flat case, the conformal factor is aligned with the potential function, which yields cohomogeneity-one symmetry.

We classify all left-invariant affine Ricci solitons associated with the Bott connections of two fixed rank-two distributions on the seven standard families of three-dimensional Lorentzian Lie groups. The distributions are defined by a fixed orthogonal change of the spacelike frame. A universal adapted-frame calculation provides explicit connection matrices, curvature operators and the symmetrized Ricci tensor. The equation in the orthogonal-complement direction forces every left-invariant solution to be steady. The remaining equations yield elementary parameter families and finitely many real algebraic branches: four normalized branches for $G_2$, six branches for each sign parameter in $G_4$, and two quartic branches in $G_7$ in addition to its elementary families. Resultants, subresultants, exact polynomial reduction and Sturm root isolation establish necessity, sufficiency and exhaustiveness. Degenerate cases are treated separately, including all families with nonunique soliton vector fields. An invertible frame transformation relates the two distribution problems. The classification is in the stated structure parameters; no quotient by isometries, automorphisms or scaling is taken.

This study develops numerical methods for computing homogeneous and isotropic cosmological models on manifolds with non-trivial spatial topologies. The use of Ricci flow is introduced here to aid in the construction of globally homogeneous initial data on these manifolds. The initial data produced in this way are evolved by solving Einstein's equation numerically to construct homogeneous and isotropic cosmological models on three exemplar manifolds in this study: Seifert-Weber space, the lens space $L(5,2)$, and Sixth-Turn space.

In the study of geometric structures on nilmanifolds, two-step nilpotent Lie algebras obtained by graphs play an important role. Recently, additional examples of nilpotent Lie algebras have been constructed from finite quivers without cycles. These latter examples can have arbitrarily high degrees of nilpotency, and admit Riemannian Ricci soliton metrics. In this paper, we study nilpotent Lie algebras obtained by finite quivers without cycles that are two-step nilpotent, and we prove that they can also be obtained by graphs. Using this relationship, we demonstrate that every two-step nilpotent Lie algebra obtained by a finite quiver without cycles admits a pseudo-Riemannian Ricci-flat metric. Additionally, we also classify these nilpotent Lie algebras that admit symplectic structures.

Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.

We fix a spacelike two-dimensional left-invariant distribution and its associated almost contact structure on a three-dimensional Lorentzian Lie group, and investigate the star-Ricci contraction of the curvature of the Bott connection. We compute the complete tensor for seven standard bracket families and determine all real parameter solutions of the general algebraic equation and its self-adjoint subclass. Using the symmetric part of the tensor, we also define a Bott affine star-Ricci soliton equation and classify all left-invariant vector field solutions. Every affine solution is steady, while two non-unimodular families admit branches with a nonzero symmetrized tensor. The proofs provide complete component equations, treat all degenerate parameter branches, and establish sufficiency by explicit substitution. We also distinguish almost contact structures from contact metric structures.

Let compact $\mathrm{RCD}(\kappa ,N)$ spaces converge in the measured Gromov–Hausdorff sense, and suppose that they have $(\delta ,\rho )$-Reifenberg local covers for one fixed $\rho >0$, with $|\kappa |{\rho }^2\le 1$ and $\delta$ small. We prove that, along a subsequence, the spaces are singular fibrations over the limit: their canonical frame spaces converge to a topological manifold with a locally linear action of $O(n)$ whose quotient is the limit; for large indices the frame spaces fibre over this manifold, by maps that commute with $O(n)$, with fibres homeomorphic to nilmanifolds; and the induced maps to the limit are, near every point, quotients of products by the isotropy group, with fibres homeomorphic to infranilmanifolds. As applications, we extend a fibration theorem of Honda and Zhang to non-smooth RCD spaces whose local covers have injectivity radius bounded below, and we show that for aspherical spaces the isotropy groups are finite, the fibre groups inject into the fundamental groups, and the limit is the quotient of a contractible manifold by a proper cocompact action; under Rong's condition $A_s$ it is a closed aspherical manifold. Three appendices treat smooth manifolds with $\mathrm{Ric}\ge -(n-1)$ and small local rewinding Reifenberg error: a uniform short-time Ricci flow with $|\mathrm{Rm}|\le \alpha /t$, also for complete non-compact data, where it is unique among complete flows with curvature at most $A/t$; a second construction with affine nilmanifold bundles; and, conditional on Nifa's construction, examples showing that rewinding volume control alone does not give continuous Gromov–Hausdorff approximations with small error.

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.

We prove harmonicity for nonshrinking algebraic generalized Ricci solitons and for all algebraic generalized Ricci solitons on solvable Lie algebras. In the solvable setting, we characterize expanding solitons by normal extensions of generalized nilsolitons, and derive a sharp rank bound.

Let $V$ be an oriented Euclidean vector space and let $\text{D}(V)=\Lambda V^{\ast }\otimes \Lambda V^{\ast }$ be the algebra of double forms. The two exterior factors carry natural factorwise exterior and interior multiplication operators and therefore two commuting Clifford actions. The central result of the paper is an intrinsic realization of these Clifford actions inside Greub's composition algebra. For $a\in V^{\ast }$ we introduce the inhomogeneous double form $${\chi }_a=((1\otimes a)-(a\otimes 1))e^g,$$ prove that the ${\chi }_a$ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas $$C_R(a)\omega ={\chi }_a\circ \omega ,C_L(a)\omega =-\omega \circ {\chi }_a.$$ For $h\in {\text{D}}^{1,1}$ this yields a canonical four-component decomposition of the coupled left–right Clifford action into exterior multiplication, double contraction and two generalized Bianchi operators. For the metric double form $g$ these four operators generate two commuting ${\mathfrak{sl}}_2$ actions. The construction extends canonically to arbitrary bidegree. In bidegree $(2,2)$ its degree-preserving component admits a direct comparison with the double-form extension of the sharp product: for every $R\in {\text{D}}^{2,2}$, $${\Gamma }_{1,1}(R)(\omega )=\frac{1}{4}R\#\omega .$$ The point of this identity is not a new definition of $R\#$, which is already constructed from Clifford commutators, but the fact that it appears canonically as the central homogeneous component of the higher left–right Clifford transform. As an application, the algebraic quadratic map $\mathcal{Q}(R)=-\frac{1}{4}R\#R$ satisfies $$\mathcal{Q}(R)=-{\Gamma }_{1,1}(R)R,$$ and its linearization at $R$ is the operator $-2{\Gamma }_{1,1}(R)$. For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.

Let $G$ be a compact connected semisimple Lie group, let $Q$ be a bi-invariant metric and let $\rho$ be its Cartan three-form. We prove that every left-invariant generalized Ricci flow with $[H_0]=[\rho ]$ exists for all positive time and converges smoothly to a Bismut-Ricci-flat pair without rescaling or pullback. The standard pair uniquely maximizes generalized scalar curvature in its torsion class, and every left-invariant Bismut-Ricci-flat pair on a compact simple group is standard, answering questions of Fusi, Lafuente and Stanfield and of Lauret and Will. On the product of two copies of a compact simple group, nonstandard stationary pairs in the Cartan class are known; we construct nonstationary flows with the exact initial form $[H_0]=[\rho ]$ and nonstandard limits, disproving the general semisimple case of the convergence conjecture of Garcia-Fernandez and Streets. The proofs use a proper nonincreasing energy and a decomposition of Killing fields that parametrizes the stationary pairs by nilpotent Lie algebra endomorphisms.

We prove new structural results for algebraic generalized Ricci solitons on solvable Lie algebras and we discuss the existence of examples in low dimensions. We establish that all non-trivial solvable algebraic generalized Ricci solitons must be expanding and give necessary conditions for the existence of such solitons on one-dimensional extensions of abelian Lie algebras. Finally, we classify algebraic generalized Ricci solitons on three-dimensional Lie algebras, and algebraic generalized Ricci solitons on four-dimensional solvable unimodular Lie algebras, up to isometry and scaling.

Let S^n+1_+ be the open hemisphere of the unit sphere S^n+1 centred at o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly convex hypersurfaces enclosing o, where r is the geodesic distance to o. We consider both the supercritical regime beta>1+k alpha and the critical regime beta=1+kalpha, where beta is the growth order of the profile at the origin, f(r) almost equals r^beta as r descends to 0. Under the structural condition that f^{1/(1+kalpha)} is convex, we prove long-time existence and preservation of strict convexity. The normalized radial function converges smoothly and exponentially to a constant: to 1 and to R_infty>0, resp. in different two cases. Thus the normalized radial graphs become round, while the original hypersurfaces contract to o.

Let H^n+1 be hyperbolic space of sectional curvature -1, with a fixed point o. We study the non-homogeneous curvature flow X_t=-f(r) sigma_k^alpha nu of smooth, closed, strictly h-convex hypersurfaces enclosing o, where r is the geodesic distance to o and alpha>0. The radial weight f is modeled on sinh^betar; we treat both regimes: beta>1+kalpha and beta=1+kalpha. Under a structural condition of f, the flow exists smoothly for all time, preserves strict h-convexity, and contracts to o. The normalized radial function converges, exponentially in normalized time: to 1 and to a positive constant R_infty respectively in two different cases.

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.

We prove that Lagrangian mean curvature flow starting from a closed, embedded, cohomogeneity-one Lagrangian in a closed, positive Kähler–Einstein manifold exists for all time, remains embedded, and converges smoothly and graphically to a minimal Lagrangian, under natural exactness and regularity assumptions. We also study the generalised Lagrangian mean curvature flow of Behrndt in Kähler manifolds which are almost-Einstein in the sense that the Ricci form satisfies $\rho =C\omega +nd{d}^{c}f$, and which satisfy $C>0$. With analogous assumptions on the flow, we obtain subconvergence to an $f$-minimal Lagrangian submanifold, with an upgrade to smooth graphical convergence in the case that $(M,g,f)$ is analytic. The proof proceeds by first reducing the flow to a weighted curve shortening flow on a compact two-dimensional orbifold. We then establish both a Grayson-type description of finite-time singularities and a long-time subconvergence theorem for weighted curve shortening flow in the orbifold setting. Finally, we use a Łojasiewicz–Simon inequality argument to upgrade to smooth convergence of the full flow.