Open questions and Problems
This document describes some open problems that I consider as interesting.
The occasion for this is a conference in Nantes in October 2012. The organizer
Gilles Carron had the good idea to collect such problems.
Bernd Ammann, Regensburg
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Problem 1: Yamabe problem and Symmetry
Joint work with M. Dahl and E. Humbert.
Let Hkc be the simply connected, complete k-dimensional Riemannian manifold with sectional curvature −c2 , i.e. it is hyperbolic space for c = 1, rescaled
hyperbolic space for 0 < c < 1 and and Euclidean space for c = 0. Let
Sn ⊂ Rn+1 be the standard sphere. We would like to better understand the
Yamabe problem on Mk,n
:= Hkc × Sn .
c
On a Riemannian manifold (M, g) of dimension m one defines the Yamabe
functional
R
a|du|2 + scal u2
FM,g (u) := M
kuk2Lp (M )
where we defined a := 4(m − 1)/(m − 2) and p := 2m/(m − 2) and where the
scalar curvature is scal. The function u is a priori in C0∞ (M ) \ {0}, where the
index 0 stands for compactly supported, but the functional FM,g extends to
non-zero u in the Sobolev space H12 (M ). The functional FM,g and its infimum
Y (M, g) :=
inf
u∈H12 (M )\{0}
FM,g (u)
play an important role in the solution of the Yamabe problem, i.e. the problem
of finding a metric of constant scalar curvature conformal to g. This problem
was solved affirmatively for compact M by Trudinger, Aubin, and Schoen.
Motivated from bordism theory, it is important to understand the behaviour
of Y (M ) under topological surgeries. And this requires good explicit lower
bounds for the invariants Y (Mk,n
c ) for c ∈ [0, 1]. It is not difficult to show
)
>
0
for
n
≥
2,
and
with considerably more effort one obtains
that Y (Mk,n
c
λk,n := inf c∈[0,1] Y (Mk,n
)
>
0.
c
These constants admit interesting applications using bordism theory.
Conjecture 1.1 The function c 7→ Y (Mk,n
c ) is increasing for n ≥ 2.
Conjecture 1.2 For n ≥ 2 and c < 1 there is a function u ∈ H12 (Hkc ) ⊂
(u) = Y (Mk,n
H12 (Mk,n
c ) such that FMk,n
c ).
c
In Conjecture 2, the space H12 (Hkc ) is viewed as those functions in H12 (Mk,n
c )
which are constant along the spheres.
Conjecture 2 does not hold for c ≥ 1. For c = 1 there are both minimizers
u which are constant along the spheres, and minimizers which are not constant
1
along the spheres. For c < 1 one expects that being non-constant along the
spheres is punished by a larger value of the functional FMk,n
and thus forbidden
c
for a minimizer. Conjecture 2 for c = 0 was already conjectured by Akutagawa,
Florit and Petean.
Assuming Conjecture 2, the calculation of Y (Mk,n
c ) relies on an ODE which
can be solved numerically. Under this assumption Conjecture 1 was verified in
many cases numerically.
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Problem 2: Yamabe problem on products
Joint work with M. Dahl and E. Humbert.
We say that a Riemannian metric g on compact manifold M is a Yamabe
metric if FM,g (1) = Y (M, g). This is equivalent to saying that Y (M, g̃) ≥
Y (M, g) for all metric g̃ conformal to g. A Yamabe metric always has constant
scalar curvature, and if Y (M, g) ≤ 0 every constant scalar curvature metric is
also a Yamabe metric. This is no longer true if Y (M, g) > 0. Now assume that
g1 resp. g2 are Yamabe metrics on M1 resp. M2 , and assume Y (Mi , gi ) > 0 for
both i = 1 and i = 2. Then the metric ht := g1 + tg2 , t > 0 on M1 × M2
has constant scalar curvature, but is no Yamabe metric for t close to 0 and for
large t.
Open Question 2.1 Is there a t such that ht is a Yamabe metric on M1 ×M2 ?
The answer is yes if both metrics gi are Einstein metrics. There is a t0 such
that ht0 is also Einstein, and Obata’s Theorem implies that it is a Yamabe
metric. Due to results by Böhm-Wang-Ziller, we then see that ht is Yamabe for
t close to t0 .
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Problem 3: Deformation of manifolds with
special holonomy
Joint work with H. Weiß and F. Witt.
We study families of metrics gt , t ∈ R, t 7→ gt continuous, on a compact spin
manifold M such that all metrics gt admit a non-trivial parallel spinor. This
implies special holonomy groups Hol(gt ) ⊂ SO(n).
Open Question 3.1 Are there families gt of such metrics such that Hol(gt1 )
is not isomorphic to Hol(gt2 ) for some t1 , t2 ∈ R?
If the answer to the question is Yes, then a finite covering of such an M
should be a product M1 × M2 , so assume M = M1 × M2 for simplicity. In
many cases we found obstructions from index theory, or from the growth type
of the fundamental group. However, it seems plausible that for example for
dim M = 14, there could be a product of two Ricci-flat Kähler manifolds such
that it admits Ricci-flat Kähler deformations which are not of product type.
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This would yield an example to the above Question. It seems to me that someone
familiar with Tian-Todorov theory should see this as an easy exercise.
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