Page 355 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 355
GEOMETRIC ANALYSIS AND LOW-DIMENSIONAL TOPOLOGY (MS-59)
Wedge theorems for ancient mean curvature flows
Niels Martin Møller, nmoller@math.ku.dk
University of Copenhagen, Denmark
We show that so-called "wedge theorems" hold for all properly immersed, not necessarily com-
pact, ancient solutions to the mean curvature flow in Rn+1. Such nonlinear parabolic Liouville-
type results add to a long story, generalizing recent results for self-translating solitons, which in
turn imply the minimal surface case (Hoffman-Meeks, ’90) that contains the classical cases of
cones (Omori ’67) and graphs (Nitsche, ’65). As an application we classify the convex hulls of
the spacetime tracks of all proper ancient flows, without any of the usual curvature assumptions.
The proofs make use of a linear parabolic Omori-Yau maximum principle for (non-compact)
ancient flows. This is joint work with F. Chini.
Alternating links, rational balls, and tilings
Brendan Owens, brendan.owens@glasgow.ac.uk
University of Glasgow, United Kingdom
Coauthor: Josh Greene
If an alternating knot is a slice of a knotted 2-sphere, then it follows from work of Greene
and Jabuka, using Donaldson’s diagonalisation combined with Heegaard Floer theory, that the
flow lattice L of its Tait graph admits an embedding in the integer lattice Zn of the same rank
which is “cubiquitous”: every unit cube in Zn contains an element of L. The same is true more
generally for an alternating link whose double branched cover bounds a rational homology 4-
ball. This results in an upper bound on the determinant of such links. In this talk I will describe
recent joint work with Josh Greene in which we classify alternating links for which this upper
bound on determinant is realised. This makes use of Minkowski’s conjecture, proved by Hajós
in 1941, which states that every lattice tiling of Rn by cubes has a pair of cubes which share a
facet.
Negatively curved Einstein metrics on quotients of 4-dimensional
hyperbolic manifolds
Bruno Premoselli, bruno.premoselli@ulb.be
Université Libre de Bruxelles, Belgium
Few examples of closed Einstein manifolds with negative scalar curvature are known in dimen-
sions larger than 4, and until recently it was believed that the only negatively curved ones were
the trivial ones, ie closed quotients of (complex)-hyperbolic space. We will construct in this
talk new examples of non-trivial closed negatively curved Einstein 4-manifolds.
More precisely, we will show that Einstein metrics with negative sectional (and scalar) cur-
vature can be found on quotients of “large” closed hyperbolic 4-manifolds with symmetries.
The proof is via a glueing procedure, starting from an approximate Einstein metric that is ob-
tained as the interpolation between a “black-hole – type” model metric near the symmetry locus
and the hyperbolic metric at large distances. This is a joint work with J. Fine (ULB, Brussels).
353
Wedge theorems for ancient mean curvature flows
Niels Martin Møller, nmoller@math.ku.dk
University of Copenhagen, Denmark
We show that so-called "wedge theorems" hold for all properly immersed, not necessarily com-
pact, ancient solutions to the mean curvature flow in Rn+1. Such nonlinear parabolic Liouville-
type results add to a long story, generalizing recent results for self-translating solitons, which in
turn imply the minimal surface case (Hoffman-Meeks, ’90) that contains the classical cases of
cones (Omori ’67) and graphs (Nitsche, ’65). As an application we classify the convex hulls of
the spacetime tracks of all proper ancient flows, without any of the usual curvature assumptions.
The proofs make use of a linear parabolic Omori-Yau maximum principle for (non-compact)
ancient flows. This is joint work with F. Chini.
Alternating links, rational balls, and tilings
Brendan Owens, brendan.owens@glasgow.ac.uk
University of Glasgow, United Kingdom
Coauthor: Josh Greene
If an alternating knot is a slice of a knotted 2-sphere, then it follows from work of Greene
and Jabuka, using Donaldson’s diagonalisation combined with Heegaard Floer theory, that the
flow lattice L of its Tait graph admits an embedding in the integer lattice Zn of the same rank
which is “cubiquitous”: every unit cube in Zn contains an element of L. The same is true more
generally for an alternating link whose double branched cover bounds a rational homology 4-
ball. This results in an upper bound on the determinant of such links. In this talk I will describe
recent joint work with Josh Greene in which we classify alternating links for which this upper
bound on determinant is realised. This makes use of Minkowski’s conjecture, proved by Hajós
in 1941, which states that every lattice tiling of Rn by cubes has a pair of cubes which share a
facet.
Negatively curved Einstein metrics on quotients of 4-dimensional
hyperbolic manifolds
Bruno Premoselli, bruno.premoselli@ulb.be
Université Libre de Bruxelles, Belgium
Few examples of closed Einstein manifolds with negative scalar curvature are known in dimen-
sions larger than 4, and until recently it was believed that the only negatively curved ones were
the trivial ones, ie closed quotients of (complex)-hyperbolic space. We will construct in this
talk new examples of non-trivial closed negatively curved Einstein 4-manifolds.
More precisely, we will show that Einstein metrics with negative sectional (and scalar) cur-
vature can be found on quotients of “large” closed hyperbolic 4-manifolds with symmetries.
The proof is via a glueing procedure, starting from an approximate Einstein metric that is ob-
tained as the interpolation between a “black-hole – type” model metric near the symmetry locus
and the hyperbolic metric at large distances. This is a joint work with J. Fine (ULB, Brussels).
353
