Page 122 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 122
TOPICS IN COMPLEX AND QUATERNIONIC GEOMETRY (MS-74)
Accordingly, the exterior derivative d splits into four operators
d : Ap,q(M ) → Ap+2,q−1(M ) ⊕ Ap+1,q(M ) ⊕ Ap,q+1(X) ⊕ Ap−1,q+2(M )
d = µ + ∂ + ∂ + µ¯ ,
where µ and µ¯ are differential operators that are linear over functions.
Let g be a Hermitian metric on (M, J). Denote by
∆∂ := ∂ ∂∗ + ∂∗∂
the ∂-Laplacian. Then ∆∂ is an elliptic differential operator. We study the space of ∂-harmonic
forms on (M, J, g). Special results are obtained for dimR M = 4. This a joint work with
Nicoletta Tardini.
On a construction of quaternionic and octonionic Riemann surfaces
Fabio Vlacci, fvlacci@units.it
University of Trieste, Italy
We present an original way to introduce quaternionic and octonionic analogs of the classical
Riemann surfaces.
The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard
Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition
of slice conformal or slice isothermal parameterization of Riemann 4-manifold and 8-manifold.
These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs
of slice regular functions, the 4 and 8 dimensional sphere and helicoidal and catenoidal mani-
folds.
This is a joint work with Graziano Gentili and Jasna Prezelj.
120
Accordingly, the exterior derivative d splits into four operators
d : Ap,q(M ) → Ap+2,q−1(M ) ⊕ Ap+1,q(M ) ⊕ Ap,q+1(X) ⊕ Ap−1,q+2(M )
d = µ + ∂ + ∂ + µ¯ ,
where µ and µ¯ are differential operators that are linear over functions.
Let g be a Hermitian metric on (M, J). Denote by
∆∂ := ∂ ∂∗ + ∂∗∂
the ∂-Laplacian. Then ∆∂ is an elliptic differential operator. We study the space of ∂-harmonic
forms on (M, J, g). Special results are obtained for dimR M = 4. This a joint work with
Nicoletta Tardini.
On a construction of quaternionic and octonionic Riemann surfaces
Fabio Vlacci, fvlacci@units.it
University of Trieste, Italy
We present an original way to introduce quaternionic and octonionic analogs of the classical
Riemann surfaces.
The construction of these manifolds has nice peculiarities and the scrutiny of Bernhard
Riemann approach to Riemann surfaces, mainly based on conformality, leads to the definition
of slice conformal or slice isothermal parameterization of Riemann 4-manifold and 8-manifold.
These new classes of manifolds include slice regular quaternionic and octonionic curves, graphs
of slice regular functions, the 4 and 8 dimensional sphere and helicoidal and catenoidal mani-
folds.
This is a joint work with Graziano Gentili and Jasna Prezelj.
120
