MOVING MONOTONICITY FORMULAE FOR MINIMAL SUBMANIFOLDS IN CONSTANT CURVATURE KEATON NAFF AND JONATHAN J. ZHU

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MOVING MONOTONICITY FORMULAE FOR MINIMAL SUBMANIFOLDS

IN CONSTANT CURVATURE

KEATON NAFF AND JONATHAN J. ZHU

Abstract. We discover new monotonicity formulae for minimal submanifolds in space forms, which

imply the sharp area bound for minimal submanifolds through a prescribed point in a geodesic ball.

These monotonicity formulae involve an energy-like integral over sets which are, in general, not

geodesic balls. In the Euclidean case, these sets reduce to the moving-centre balls introduced by

the second author in [Zhu18].

1. Introduction

Recently, in [NZ22], we studied an area estimate for minimal submanifolds in geodesic balls

in space forms which pass through a prescribed point. Such area estimates sometimes follow

from a suitable monotonicity formula; monotonicity formulae naturally encapsulate strictly more

information and are often used to give more precise control of geometric quantities. In the following

brief note, we will exhibit some new monotonicity formulae for minimal submanifolds in space forms.

These formulae can be used to recover the sharp area estimates proved in [NZ22] and [BH17].

Consider a space form M∈ {Hn,Rn,Sn}, and a geodesic ball Bn

Rin Mwith radius R∈

(0,1

2diam(M)) and centre o. Deﬁne A(r) := Rr

0sn(t)k−1dt, where g=dr2+ sn(r)2gSn−1and

sn(r) is the usual warping function. Given a point yin Bn

R, let γ⊂Mbe the maximal geodesic

containing oand y. There is a foliation of Bn

Rby totally geodesic (n−1)-dimensional disks Γs⊂Bn

s∈(−R, R), which meet γorthogonally.1Let rybe the distance function from the point y. There

is a unique function uson Bn

Rwhich agrees with ryon ∂Bn

Rand such that us=u(s) is constant

on each Γs(see Sections 2.4 and 3.2 in [NZ22] for more details and Deﬁnition 4.2 below).

Theorem 1.1. Suppose M∈ {Hn,Rn,Sn}is a space form, R∈(0,1

2diam(M)), and y∈Bn

R. If

M=Sn, further assume that cos(R+d(0, y))2≥2

k. Suppose that Σis a k-dimensional minimal

submanifold in Bn

Rwith ∂Σ⊂∂Bn

Deﬁne f:= A(ry)

A(us)and let Et:= {f≤t}. Let Σ0be any totally geodesic k-disk orthogonal to

the geodesic γcontaining oand y. There is a continuous family of functions Gt≥0on Bn

R, with

G0= 0, so that the quantity

Q(t) := RΣ∩Et|∇>ry|2−Gt|∇>u|2

|Σ0∩Et|

is monotone increasing for t∈[0,1]. Moreover, Qis constant if and only if Σis a totally geodesic

k-disk orthogonal to γ.

For a more precise statement, the reader may consult Theorem 4.4, which also includes certain

other monotonicity formulae for M∈ {Hn,Rn}(see also Remark 4.3). One of these is equivalent

to the moving-centre monotonicity formula previously found by the second author [Zhu18]. This

1In Rn, for instance, these are just the intersection of hyperplanes orthogonal to ywith Bn

arXiv:2210.03263v1 [math.DG] 7 Oct 2022

2 KEATON NAFF AND JONATHAN J. ZHU

latter monotonicity formula is a proper (unweighted) area monotonicity, whereas all of the new

monotonicity formulae in this paper require a weight similar to the one in Theorem 1.1. We note

that even in the classical setting, y=o, it is not known whether an area-monotonicity holds in

the sphere. Nevertheless, there is a very natural weighted monotonicity (Theorem 3.3) that holds

for every space form. Theorem 1.1 can be thought of as a generalisation of this classical weighted

monotonicity. There is also a related boundary monotonicity that holds in the classical setting

(Theorem 3.4), but we do not know if an analogue holds here. We remark that there seem to be

fewer settings in extrinsic geometry where monotonicity formulae are known to hold, compared

with the intrinsic setting (where, for instance, versions of the Bishop-Gromov monotonicity are

known to hold under a variety of general settings).

In all of our monotonicity formulae, the integrand in the numerator is always bounded above

by 1, and converges to 1 as f→0. For small tthe sublevel sets Etapproximate geodesic balls

around y. When M∈ {Hn,Rn}, we have E1=Bn

R. Thus, one may deduce the sharp area bound

|Σ|≥|Σ0∩Bn

R|if Σ passes through the prescribed point y(see Corollary 4.7).

The sets Etbehave slightly diﬀerently if M=Snand in general we only have E1=Bn

R∩Bn

2(y).

The domain Bn

R∩Bn

2(y) also appeared as an obstruction to the vector ﬁeld approach used in

[NZ22] to prove the sharp area estimate in Snfor certain values of Rand y(see Section 5 there for

more discussion). In our main theorem, we are only able to establish the monotonicity of Q(t) if

cos(R+d(o, y))2≥2

kand this implies that Bn

R⊂Bn

2(y), hence E1=Bn

R. Note that this condition

means the monotonicity does not hold for any Rand yif k= 2. When the monotonicity does

hold, we can also deduce the sharp area estimate in the sphere. We note that the direct vector ﬁeld

method in [NZ22] gave the sharp area estimate under somewhat more general conditions on R, y.

The proof of Theorem 1.1 originates from ideas from [Zhu18], but with two key conceptual

realisations: that the moving-centre balls should be replaced by a suitable family of sublevel sets,

and that the area should be replaced by a suitable weight. These developments give rise to several

conditions that needs to be delicately balanced against each other. First, we extend the notion of

moving-centre balls to the sublevel sets of f=A(ry)

A(us). This is motivated by the vector ﬁeld used to

solve the prescribed point problem in [NZ22] as well as the fact that it agrees with the known case

in [Zhu18]. Partly motivated by the classical weighted monotonicity which holds in all space forms,

we also introduce a weight wt, which we additionally allow to depend on t. A monotonicity for the

energy-like quantity RΣ∩Etwtfollows so long as one can ﬁnd a family of vector ﬁelds Wtsatisfying:

(1) (boundary condition) hWt,∇>f

fi ≤ wton Σ ∩∂Et

(2) (divergence condition) divΣ(Wt)≥wt−t∂twton Σ ∩Et.

(See Lemma 4.1). In order to deduce the desired sharp area estimates from this monotonicity, the

weight wtand the sets Etmust satisfy a number of additional constraints. Most importantly, we

should have that:

•wt≤1 (with equality on the totally geodesic disks orthogonal to γ);

•wt→1 as t→0;

•E1=Bn

•Etapproximates balls about yas t→0.

One of the main diﬃculties in discovering a suitable monotonicity formula is to simultaneously

produce the three families Et, wt, Wt, which satisfy the interdependent conditions above.

MOVING MONOTONICITY FORMULAE IN CONSTANT CURVATURE 3

Acknowledgements. KN was supported by the National Science Foundation under grant DMS-

2103265.

2. Preliminaries

In this section, we will very brieﬂy review some preliminaries necessary for the proof of the new

monotonicity formulae. The reader can more details in [NZ22].

In what follows, M∈ {Hn,Rn,Sn}is one of the space forms. We let Bn

t=Bn

t(o) denote the

geodesic ball of radius taround a ﬁxed point o∈M, which we call the origin. We ﬁx some

R∈(0,1

2diam(M)) and some y∈Bn

R, and let Σ ⊂Bn

Rdenote a k-dimensional (smooth) minimal

submanifold which passes through yand satisﬁes ∂Σ⊂∂Bn

R. Deﬁne

(2.1) sn(r) = 









sinh(r), M =Hn

r, M =Rn

sin(r), M =Sn

and set cs(r) := sn0(r), as well as tn(r) := sn(r)/cs(r) and ct(r) = 1/tn(r).

Given z∈M, we introduce the shorthand rz(x) = d(x, z) for the distance function on M. Away

from zand the cut locus of zthe function rzis smooth. For our origin o∈M, we will write r(x)

in place of ro(x).

As in [NZ22] we deﬁne a radius r(y) by

(2.2) r(y) := (cs−1cs(R)

cs(r(y)) , M ∈ {Hn,Sn}

(R2−r(y)2)1

2, M =Rn.

If r(y) = 0, we understand that r(y) = R.

For r∈[0,1

2diam(M)), we recall from the introduction the deﬁnition

(2.3) A(r) := Zr

sn(t)k−1dt.

Observe that A(r) is positive and increasing. Moreover,

(2.4) A00(r)

A0(r)= (k−1) ct(r).

The area of a k-dimensional totally geodesic disk in the space form Mis

(2.5) |Bk

r|=A(r)|Sk−1|.

Finally, we note that as r→0, these functions have the asymptotics

(2.6) A0(r) = rk−1+o(rk−1), A(r) = 1

krk+o(rk).

2.1. The functions sand ρ.As in the introduction, let γ⊂Mbe the unique maximal geodesic

containing the points oand y. Let ρ(x) := infz∈γd(x, z) denote the distance to the geodesic γ. For

each point xwith ρ(x)<1

2diam(M), there exists a unique point zx∈γsuch that ρ(x) = d(x, zx).

Note that diam(M)<∞only when M=Sn. In this case {ρ(x) = 1

2diam(M)}consists of a copy of

Sn−2and we let o0denote the antipodal point to o. For M=Sn, we set E={ρ=1

2diam(M)}∪{x∈

M:ρ(x)<1

2diam(M) and zx=o0}. Otherwise, we take E=∅. Note in particular that Bn

R⊂M\E

whenever R < 1

2diam(M). Now, deﬁne sign : γ\ E → {−1,0,1}by sign(z) = 1 if z∈γ\ E lies on

the same side of oas y, sign(o) = 0, and sign(z) = −1 otherwise.

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MOVINGMONOTONICITYFORMULAEFORMINIMALSUBMANIFOLDSINCONSTANTCURVATUREKEATONNAFFANDJONATHANJ.ZHUAbstract.Wediscovernewmonotonicityformulaeforminimalsubmanifoldsinspaceforms,whichimplythesharpareaboundforminimalsubmanifoldsthroughaprescribedpointinageodesicball.Thesemonotonicityformulaeinvolveanenergy-l...

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MOVING MONOTONICITY FORMULAE FOR MINIMAL SUBMANIFOLDS IN CONSTANT CURVATURE KEATON NAFF AND JONATHAN J. ZHU

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