Searcharxiv⌕ Search

arXiv subjects

Filippo Salis

Publications and source records attributed to Filippo Salis.

16 recordsLinked to original sources
arXiv2026

Sophus Lie's problem on two-dimensional metrics with projective symmetries: completing the local classification

We complete the local classification, up to isometry, of $2$-dimensional pseudo-Riemannian metrics (i.e. both Riemannian and Lorentzian), admitting a projective symmetry algebra of dimension at least two. The new contribution is the treatment of the non-regular case: we obtain a complete list of mutually non-isometric normal forms in neighbourhoods of points where the action of the projective Lie symmetry algebra fails to be regular, including all singular behaviours that can occur. Together with the known classification in the regular case, this completes the solution of the problem posed by Sophus Lie in 1882.

math.DG↗
arXiv2026

Projectively equivalent para-Kaehler and para-Kaehler-Einstein metrics with non-parallel Benenti tensors and their normal forms in dimension four

The study of projectively equivalent metrics, i.e., metrics sharing the same unparametrized geodesics, is a classical and well-established area of investigation. In the Kaehler context, such branch of research goes by the name of c-projective geometry: it mainly studies c-projectively equivalent metrics, i.e., Kaehler metrics sharing the same curves that are the complex analogue of the geodesics, called J-planar, where J is the complex structure. In this paper, we develop the theory of the projective equivalence in the para-Kaehler context by studying para-Kaehler metrics sharing the same T-planar curves, where T is the para-complex structure: we call such metrics pc-projectively equivalent. After establishing some general results in arbitrary dimension, we focus on the 4-dimensional case. One of the main achievement is a local description of 4-dimensional pc-projectively (but not affinely) equivalent metrics and, as an application of this result, we characterize which of them are of Einstein type.

math.DG↗
arXiv2025

Toric para-Kaehler-Einstein manifolds immersed in para-Kaehler space forms

A classical and long-staying problem addressed, among others, by Calabi and Chern, is that to find a complete list of mutually non-isometric Kaehler-Einstein manifolds immersed in a finite-dimensional Kaehler space form. We address the same problem in the para-Kaehler context and, then, we find a list of mutually non-isometric toric para-Kaehler manifolds analytically immersed in a finite-dimensional para-Kaehler space form

math.DG↗
arXiv2025

Para-Kaehler immersions in para-Kaehler space forms

In this paper, we provide necessary and sufficient conditions for the existence of para-Kaehler immersions in para-Kaehler space forms. As a consequence, we prove that, in general, a local para-Kaehler immersion cannot be globally extended, even if it is defined on a simply connected para-Kaehler manifold. Finally, we classify para-Kaehler immersions between para-Kaehler space forms.

math.DG↗
arXiv2022

Lower dimensional $S^1$-invariant Kähler-Einstein metrics via integrable structures

We focus on the classical open problem of the classification of Kähler-Einstein manifolds that can be Kähler immersed into a complex projective space endowed with the Fubini-Study metric. In particular, we will deal with such problem in the special case of Kähler- Einstein metrics admitting symmetries of rotational type. This leads to certain integrable distributions allowing a classification of such metrics.

math.DG↗
arXiv2021

On canonical radial Kaehler metrics

We prove that a radial Kaehler metric g is Kaehler-Einstein if and only if one of the following conditions is satisfied: 1. g is extremal and it is associated to a Kaehler-Ricci soliton; 2. two different generalized scalar curvatures of g are constant; 3. g is extremal (not cscK) and one of its generalized scalar curvature is constant.

math.DG↗
arXiv2021

Kaehler Ricci solitons induced by infinite dimensional complex space forms

We exhibit families of non trivial (i.e. not Kaehler-Einstein) radial Kaehler-Ricci solitons (KRS), both complete and not complete, which can be Kaehler immersed into infinite dimensional complex space forms. This result shows that the triviality of a KRS induced by a finite dimensional complex space form proved in [12] does not hold when the ambient space is allowed to be infinite dimensional. Moreover, we show that the radial potential of a radial KRS induced by a non-elliptic complex space form is necessarily defined at the origin.

math.DG↗
arXiv2020

Extremal Kaehler metrics induced by finite or infinite dimensional complex space forms

In this paper we address the problem of studying those complex manifolds $M$ equipped with extremal metrics $g$ induced by finite or infinite dimensional complex space forms. We prove that when $g$ is assumed to be radial and the ambient space is finite dimensional then $(M, g)$ is itself a complex space form. We extend this result to the infinite dimensional setting by imposing the strongest assumption that the metric $g$ has constant scalar curvature and is well-behaved (see Definition 1 in the Introduction). Finally, we analyze the radial Kaehler-Einstein metrics induced by infinite dimensional elliptic complex space forms and we show that if such a metric is assumed to satisfy a stability condition then it is forced to have constant non-positive holomorphic sectional curvature.

math.DG↗
arXiv2020

On the Cauchy-Riemann geometry of transversal curves in the 3-sphere

Let $\mathrm S^3$ be the unit sphere of $\mathbb C^2$ with its standard Cauchy-Riemann (CR) structure. This paper investigates the CR geometry of curves in $\mathrm S^3$ which are transversal to the contact distribution, using the local CR invariants of $\mathrm S^3$. More specifically, the focus is on the CR geometry of transversal knots. Four global invariants of transversal knots are considered: the phase anomaly, the CR spin, the Maslov index, and the CR self-linking number. The interplay between these invariants and the Bennequin number of a knot are discussed. Next, the simplest CR invariant variational problem for generic transversal curves is considered and its closed critical curves are studied.

math.DG↗
arXiv2020

The Cauchy-Riemann strain functional for Legendrian curves in the 3-sphere

The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied and its Euler-Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature are characterized. We prove that their cr-equivalence classes are in one-to-one correspondence with the rational points of a connected planar domain. A procedure to explicitly build all such curves is described. In addition, a geometrical interpretation of the rational parameters in terms of three phenomenological invariants is given.

math.DG↗
arXiv2019

A characterization of complex space forms via Laplace operators

Inspired by the work of Z. Lu and G. Tian \cite{lutian}, in this paper we address the problem of studying those \K\ manifolds satisfying the $Δ$-property, i.e. such that on a neighborhood of each of its points the $k$-th power of the \K Laplacian is a polynomial function of the complex Euclidean Laplacian, for all positive integer $k$ (see below for its definition). We prove two results: 1. if a \K\ manifold satisfies the $Δ$-property then its curvature tensor is parallel; 2. if an Hermitian symmetric space of classical type satisfies the $Δ$-property then it is a complex space form (namely it has constant holomorphic sectional curvature). In view of these results we believe that if a complete and simply-connected \K\ manifold satisfies the $Δ$-property then it is a complex space form.

math.DG↗
arXiv2017

Two conjectures on Ricci-flat Kaehler metrics

We propose two conjectures about Ricci-flat metrics: Conjecture 1: A Ricci-flat projectively induced metric is flat. Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat. We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the metric is radial and stable-projectively induced and Conjecture 2 (see Theorem 1.2) for complex surfaces whose metric is either radial or complete and ALE. We end the paper by showing, by means of the Simanca metric, that the assumption of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to scalar-flatness (see Theorem 1.3).

math.DG↗
arXiv2016

Projectively induced rotation invariant Kähler metrics

We classify Kähler-Einstein manifolds which admit a Kähler immersion into a finite dimensional complex projective space endowed with the Fubini-Study metric, whose codimention is not greater than 3 and whose metric is rotation invariant.

math.DG↗