Conformal geometry on a class of embedded hypersurfaces in spacetimes

In this work, we study various geometric properties of embedded spacelike hypersurfaces in $1+1+2$ decomposed spacetimes with a preferred spatial direction, denoted $e^{\mu}$, which are orthogonal to the fluid flow velocity of the spacetime and admit a proper conformal transformation. To ensure non-vanishing and positivity of the scalar curvature of the induced metric on the hypersurface, we impose that the scalar curvature of the conformal metric is non-negative and that the associated conformal factor $\varphi$ satisfies $\hat{\varphi}^2+2\hat{\hat{\varphi}}>0$, where \hat{\ast} denotes derivative along the preferred spatial direction. Firstly, it is demonstrated that such hypersurface is either of Einstein type or the spatial twist vanishes on them, and that the scalar curvature of the induced metric is constant. It is then proved that if the hypersurface is compact and of Einstein type and admits a proper conformal transformation, then these hypersurfaces must be isomorphic to the 3-sphere, where we make use of some well known results on Riemannian manifolds admitting conformal transformations. If the hypersurface is not of Einstein type and have nowhere vanishing sheet expansion, we show that this conclusion fails. However, with the additional conditions that the scalar curvatures of the induced metric and the conformal metric coincide, the associated conformal factor is strictly negative and the third and higher order derivatives of the conformal factor vanish, the conclusion that the hypersurface is isomorphic to the 3-sphere follows. Furthermore, additional results are obtained under the conditions that the scalar curvature of a metric conformal to the induced metric is also constant. Finally, we consider some of our results in the context of locally rotationally symmetric spacetimes and show that, if the hypersurfaces are compact and not of Einstein type, then under certain specified conditions the hypersurface is isomorphic to the 3-sphere, where we constructed explicit examples of several proper conformal Killing vector fields along $e^{\mu}$.

Integrable cosmological models with an additional scalar field

We consider modified gravity cosmological models that can be transformed into two-field chiral cosmological models by the conformal metric transformation. For the $$R^2$$ R 2 gravity model with an additional scalar field and the corresponding two-field model with the cosmological constant and nonstandard kinetic part of the action, the general solutions have been obtained in the spatially flat FLRW metric. We analyze the correspondence of the cosmic time solutions obtained and different possible evolutions of the Hubble parameters in the Einstein and Jordan frames.

A structure of the solenoidal 2D vector and 2-tensor fields given in a domain with the conformal Riemannian metric

The Helmholtz decomposition of a vector field on potential and solenoidal parts is much more natural from physical and geometric points of view then representations through the components of the vector in the Cartesian coordinate system of Euclidean space. The structure, representation through potentials and detailed decomposition for 2D symmetric m-tensor fields in a case of the Euclidean metric is known. For the Riemannian metrics similar results are known for vector fields. We investigate the properties of the solenoidal vector and 2-tensor two-dimensional fields given in the Riemannian domain with the conformal metric and establish the connections between the fields and metrics.

Metrics on a Surface with Bounded Total Curvature

Let $g=e^{2u}g_{euc}$ be a conformal metric defined on the unit disk of ${{\mathbb{C}}}$. We give an estimate of $\|\nabla u\|_{L^{2,\infty }(D_{\frac{1}{2}})}$ when $\|K(g)\|_{L^1}$ is small and $\frac{\mu (B_r^g(z),g)}{\pi r^2}<\Lambda $ for any $r$ and $z\in D_{\frac{3}{4}}$. Then we use this estimate to study the Gromov–Hausdorff convergence of a conformal metric sequence with bounded $\|K\|_{L^1}$ and give some applications.

Chern-Yamabe problem and Chern-Yamabe soliton

Let [Formula: see text] be a compact complex manifold of complex dimension [Formula: see text] endowed with a Hermitian metric [Formula: see text]. The Chern-Yamabe problem is to find a conformal metric of [Formula: see text] such that its Chern scalar curvature is constant. In this paper, we prove that the solution to the Chern-Yamabe problem is unique under some conditions. On the other hand, we obtain some results related to the Chern-Yamabe soliton.

Determinant of the Laplacian on Tori of Constant Positive Curvature with one Conical Point

We find an explicit expression for the zeta-regularized determinant of (the Friedrichs extensions of) the Laplacians on a compact Riemann surface of genus one with conformal metric of curvature $1$ having a single conical singularity of angle $4\unicode[STIX]{x1D70B}$ .

Prescribed mean curvature equation on the unit ball in the presence of reflection or rotation symmetry

Using the flow method, we prove some existence results for the problem of prescribing the mean curvature on the unit ball. More precisely, we prove that there exists a conformal metric on the unit ball such that its mean curvature is f, when f possesses certain reflection or rotation symmetry.