Equivariant Yamabe problem with boundary

As a generalization of the Yamabe problem, Hebey and Vaugon considered the equivariant Yamabe problem: for a subgroup G of the isometry group, find a G-invariant metric whose scalar curvature is constant in a given conformal class. In this paper, we study the equivariant Yamabe problem with boundary.

Second-Order PDEs in 3D with Einstein–Weyl Conformal Structure

Einstein–Weyl geometry is a triple $$({\mathbb {D}},g,\omega )$$ ( D , g , ω ) where $${\mathbb {D}}$$ D is a symmetric connection, [g] is a conformal structure and $$\omega $$ ω is a covector such that $$\bullet $$ ∙ connection $${\mathbb {D}}$$ D preserves the conformal class [g], that is, $${\mathbb {D}}g=\omega g$$ D g = ω g ; $$\bullet $$ ∙ trace-free part of the symmetrised Ricci tensor of $${\mathbb {D}}$$ D vanishes. Three-dimensional Einstein–Weyl structures naturally arise on solutions of second-order dispersionless integrable PDEs in 3D. In this context, [g] coincides with the characteristic conformal structure and is therefore uniquely determined by the equation. On the contrary, covector $$\omega $$ ω is a somewhat more mysterious object, recovered from the Einstein–Weyl conditions. We demonstrate that, for generic second-order PDEs (for instance, for all equations not of Monge–Ampère type), the covector $$\omega $$ ω is also expressible in terms of the equation, thus providing an efficient ‘dispersionless integrability test’. The knowledge of g and $$\omega $$ ω provides a dispersionless Lax pair by an explicit formula which is apparently new. Some partial classification results of PDEs with Einstein–Weyl characteristic conformal structure are obtained. A rigidity conjecture is proposed according to which for any generic second-order PDE with Einstein–Weyl property, all dependence on the 1-jet variables can be eliminated via a suitable contact transformation.

Scalar curvature and the multiconformal class of a direct product Riemannian manifold

For a closed, connected direct product Riemannian manifold $$(M, g)=(M_1, g_1) \times \cdots \times (M_l, g_l)$$ ( M , g ) = ( M 1 , g 1 ) × ⋯ × ( M l , g l ) , we define its multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] as the totality $$\{f_1^2g_1\oplus \cdots \oplus f_l^2g_l\}$$ { f 1 2 g 1 ⊕ ⋯ ⊕ f l 2 g l } of all Riemannian metrics obtained from multiplying the metric $$g_i$$ g i of each factor $$M_i$$ M i by a positive function $$f_i$$ f i on the total space M. A multiconformal class $$ [\![ g ]\!]$$ [ [ g ] ] contains not only all warped product type deformations of g but also the whole conformal class $$[\tilde{g}]$$ [ g ~ ] of every $$\tilde{g}\in [\![ g ]\!]$$ g ~ ∈ [ [ g ] ] . In this article, we prove that $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of positive scalar curvature if and only if the conformal class of some factor $$(M_i, g_i)$$ ( M i , g i ) does, under the technical assumption $$\dim M_i\ge 2$$ dim M i ≥ 2 . We also show that, even in the case where every factor $$(M_i, g_i)$$ ( M i , g i ) has positive scalar curvature, $$ [\![ g ]\!]$$ [ [ g ] ] contains a metric of scalar curvature constantly equal to $$-1$$ - 1 and with arbitrarily large volume, provided $$l\ge 2$$ l ≥ 2 and $$\dim M\ge 3$$ dim M ≥ 3 .

Isothermic constrained Willmore tori in 3-space

We show that the homogeneous and the 2-lobe Delaunay tori in the 3-sphere provide the only isothermic constrained Willmore tori in 3-space with Willmore energy below $$8\pi $$ 8 π . In particular, every constrained Willmore torus with Willmore energy below $$8\pi $$ 8 π and non-rectangular conformal class is non-degenerated.

Metric topology on the moduli space

<p>We define the smooth Lipschitz topology on the moduli space and show that each conformal class is dense in the moduli space endowed with Gromov-Hausdorff topology, which offers an answer to Tuschmann’s question.</p>

An Obata-type theorem on compact Einstein manifolds with boundary

We show a kind of Obata-type theorem on a compact Einstein n-manifold $$(W, \bar{g})$$ ( W , g ¯ ) with smooth boundary $$\partial W$$ ∂ W . Assume that the boundary $$\partial W$$ ∂ W is minimal in $$(W, \bar{g})$$ ( W , g ¯ ) . If $$(\partial W, \bar{g}|_{\partial W})$$ ( ∂ W , g ¯ | ∂ W ) is not conformally diffeomorphic to $$(S^{n-1}, g_S)$$ ( S n - 1 , g S ) , then for any Einstein metric $$\check{g} \in [\bar{g}]$$ g ˇ ∈ [ g ¯ ] with the minimal boundary condition, we have that, up to rescaling, $$\check{g} = \bar{g}$$ g ˇ = g ¯ . Here, $$g_S$$ g S and $$[\bar{g}]$$ [ g ¯ ] denote respectively the standard round metric on the $$(n-1)$$ ( n - 1 ) -sphere $$S^{n-1}$$ S n - 1 and the conformal class of $$\bar{g}$$ g ¯ . Moreover, if we assume that $$\partial W \subset (W, \bar{g})$$ ∂ W ⊂ ( W , g ¯ ) is totally geodesic, we also show a Gursky-Han type inequality for the relative Yamabe constant of $$(W, \partial W, [\bar{g}])$$ ( W , ∂ W , [ g ¯ ] ) .

Constant Q-curvature metrics on conic 4-manifolds

We consider the constant Q-curvature metric problem in a given conformal class on a conic 4-manifold and study related differential equations. We define subcritical, critical, and supercritical conic 4-manifolds. Following [M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 1991, 2, 793–821] and [S.-Y. A. Chang and P. C. Yang, Extremal metrics of zeta function determinants on 4-manifolds, Ann. of Math. (2) 142 1995, 1, 171–212], we prove the existence of constant Q-curvature metrics in the subcritical case. For conic 4-spheres with two singular points, we prove the uniqueness in critical cases and nonexistence in supercritical cases. We also give the asymptotic expansion of the corresponding PDE near isolated singularities.

A calculus for conformal hypersurfaces and new higher Willmore energy functionals

The invariant theory for conformal hypersurfaces is studied by treating these as the conformal infinity of a conformally compact manifold. Recently it has been shown how, given a conformal hypersurface embedding, a distinguished ambient metric is found (within its conformal class) by solving a singular version of the Yamabe problem [21]. This enables a route to proliferating conformal hypersurface invariants. The aim of this work is to give a self contained and explicit treatment of the calculus and identities required to use this machinery in practice. In addition we show how to compute the solution’s asymptotics. We also develop the calculus for explicitly constructing the conformal hypersurface invariant differential operators discovered in [21] and in particular how to compute extrinsically coupled analogues of conformal Laplacian powers. Our methods also enable the study of integrated conformal hypersurface invariants and their functional variations. As a main application we prove that a class of energy functions proposed in a recent work have the right properties to be deemed higher-dimensional analogues of the Willmore energy. This complements recent progress on the existence and construction of different functionals in [22] and [20].