J. Evol. Equ.
© 2018 Springer International Publishing AG,
part of Springer Nature
Journal of Evolution
The Yamabe ﬂow on incomplete manifolds
Abstract. This article is concerned with developing an analytic theory for second-order nonlinear parabolic
equations on singular manifolds. Existence and uniqueness of solutions in an L
-framework are established
by maximal regularity tools. These techniques are applied to the Yamabe ﬂow. It is proven that the Yamabe
ﬂow admits a unique local solution within a class of incomplete initial metrics.
Nowadays, there is a rising interest in the study of differential operators on mani-
folds with singularities, which is motivated by a variety of applications from applied
mathematics, geometry and topology. All the work is more or less related to the semi-
nal paper by Kondrat’ev . Among the tremendous amount of the literature on this
topic, I would like to mention two lines of research on the study of differential oper-
ators of Fuchs type, which have been introduced independently by Melrose [36,37],
Nazaikinskii et al. , Schulze [45,46] and Schulze and Seiler .
One important direction of research on singular analysis is connected with the
so-called b-calculus and its generalizations on manifolds with cylindrical ends. See
[36,37]. Many authors have been very active in this direction.
Research along another line, known as cone differential operators, has also been
known for a long time. Operators in this line of research are modeled on conical
manifolds. During the recent decade, many mathematicians have applied analytic
tools like bounded imaginary powers, H
-calculus and R-sectoriality, c.f. Sect. 4.1
for precise deﬁnitions, to study the closed extensions of cone differential operators
in Mellin–Sobolev spaces and to investigate many interesting nonlinear parabolic
problems on conical manifolds. See for instance [13,42–44]. A comparison between
the b-calculus and the cone algebra can be found in .
There has been more recent progress in understanding elliptic operators on mani-
folds with higher-order singularities, e.g., manifolds with edge ends. The reader may
refer to [7,30–33,46–48] for more details. The amount of research on differential
Mathematics Subject Classiﬁcation: 35K55, 35K67, 35R01, 53C21, 53C44, 58J99
Keywords: Singular parabolic equations, Maximal L
-regularity, Incomplete Riemannian manifolds,
Geometric evolution equations, The Yamabe ﬂow.