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Dissertation

zur Erlangung des Grades eines Doktors der Naturwissenschaften

der Fakultat fur Mathematik und Physik¨ ¨

der Eberhard-Karls-Universitat Tubingen¨ ¨

vorgelegt von

Ulrich Menne

aus Frankfurt am Main

2008Tag der mun¨ dlichen Qualiﬁkation: 08.08.2008

Dekan: Prof. Dr. Nils Schopohl

1. Berichterstatter: Prof. Dr. Reiner Schatzle¨

2. Berich Prof. Dr. Tom IlmanenContents

Zusammenfassung in deutscher Sprache 1

Introduction 1

1 Approximation of integral varifolds 5

2 A Sobolev Poincar´e type inequality for integral varifolds 25

3 About the signiﬁcance of the 1 tilt 34

A The Isoperimetric Inequality and its applications 36

B A diﬀerentiation theorem 42

C An example concerning tilt and height decays of integral vari-

folds 44

D Elementary properties of Q valued functions 47

References 52

iiZusammenfassung in deutscher Sprache

n+mIn dieser Arbeit werden integrale n Varifaltigkeiten in R betrachtet, wel-

pche eine Bedingung an die verallgemeinerte mittlere Krummung inL -Raumen¨ ¨

erfullen. Genauer wird der Zusammenhang von Großen, welche den klassischen¨ ¨

Tilt- und Height-Excess umfassen und verallgemeinern, untersucht, insbesonde-

2re im Hinblick auf die Frage moglic¨ herC -Rektiﬁzierbarkeit solcher Varifaltigkei-

ten. Das Hauptresultat besagt, daß die Abweichung der integralen Varifaltigkeit

von einer eventuell mehrwertigen Ebene (Height-Excess) durch die Abweichung

der approximativen Tangentialraume¨ der integralen Varifaltigkeit von besagter

Ebene (Tilt-Excess) und die mittlere Krumm¨ ung kontrolliert werden kann.

Introduction

2 n+mThis work is concerned with C rectiﬁability of integral n varifolds in R ,

m,n ∈ N which are of locally bounded ﬁrst variation. More precisely, given

2assumptions on the mean curvature, the relationship between C rectiﬁability

and decay of height or tilt quantities is examined.

First,somedeﬁnitionswillberecalled. Supposethroughouttheintroduction

n+mthatm,nareasaboveandU isanonempty,opensubsetofR . Using[Sim83,

Theorem 11.8] as a deﬁnition, μ is a rectiﬁable [an integral] n varifold in U if

andonlyifμisaRadonmeasureonU andforμalmostallx∈U thereexistsan

napproximate tangent planeT μ∈G(n+m,n) with multiplicityθ (μ,x) ofμ atx

nx [andθ (μ,x)∈N],G(n+m,n) denoting the set ofn dimensional, unoriented

n+mplanes inR . The distributional ﬁrst variation of mass of μ equals

R

1 n+m(δμ)(η) = div ηdμ whenever η∈C (U,R )μ c

where div η(x) is the trace of Dη(x) with respect to T μ. kδμk denotes theμ x

total variation measure associated to δμ and μ is said to be of locally bounded

ﬁrst variation if and only if kδμk is a Radon measure. The tilt-excess and the

height-excess of μ are deﬁned by

R

−n 2tiltex (x,%,T) :=% |T μ−T| dμ(ξ),μ ξB (x)%

R

−n−2 2heightex (x,%,T) :=% dist(ξ−x,T) dμ(ξ)μ B (x)%

n+mwhenever x ∈ R , 0 < % < ∞, B (x) ⊂ U, T ∈ G(n+m,n); here S ∈%

n+mG(n+m,n) is identiﬁed with the orthogonal projection ofR ontoS and|·|

n+m n+mdenotes the norm induced by the usual inner product on Hom(R ,R ).

From the above deﬁnition of a rectiﬁablen varifoldμ one obtains thatμ almost

all of U is covered by a countable collection of n dimensional submanifolds of

n+m 1

R of class C . This concept is extended to higher orders of diﬀerentiability

by adapting a deﬁnition of Anzellotti and Serapioni in [AS94] as follows: A

k,α krectiﬁable n varifold μ in U is called countably rectiﬁable of class C [C ],

k ∈ N, 0 < α ≤ 1, if and only if there exists a countable collection of n

n+m k,α kdimensional submanifolds ofR of class C [C ] covering μ almost all of U.

k,α kThroughout the introduction this will be abbreviated to C [C ] rectiﬁability.

k,1 k+1Note that C rectiﬁability and C rectiﬁability agree by [Fed69, 3.1.15].

Decays of tilt-excess or height-excess have been successfully used in [All72,

2Bra78, Sch04a, Sch04b]. The link toC rectiﬁability is provided in [Sch04b], see

1below. In order to explain some of these results, a mean curvature condition is

introduced. AnintegralnvarifoldinU issaidtosatisfy(H ), 1≤p≤∞, ifandp

p n+m~only if either p > 1 and for some H ∈ L (μ,R ), called the generalisedμ loc

mean curvature of μ,

R

1 n+m~(δμ)(η) =− H •ηdμ whenever η∈C (U,R ) (H )μ pc

or p = 1 and

μ is of locally bounded ﬁrst variation; (H )1

n+mhere • denotes the usual inner product onR . Brakke has shown in [Bra78,

5.7] that

tiltex (x,%,T μ) =o (%), heightex (x,%,T μ) =o (%) as %↓ 0μ x x x xμ

for μ almost every x∈U provided μ satisﬁes (H ) and1

2−ε 2−εtiltex (x,%,T μ) =o (% ), heightex (x,%,T ) =o (% ) as %↓ 0μ x x x xμ

for every ε > 0 for μ almost every x ∈ U provided μ satisﬁes (H ). In case2

of codimension 1 and p > n Sch¨atzle has proved the following result yielding

optimal decay rates.

Theorem 5.1 in [Sch04a]. If m = 1, p > n, p ≥ 2, and μ is an integral n

varifold in U satisfying (H ), thenp

2 2tiltex (x,%,T μ) =O (% ), heightex (x,%,T μ) =O (% ) as %↓ 0μ x x x xμ

for μ almost all x∈U.

The importance of the improvement from 2−ε to 2 stems mainly from the

fact that the quadratic decay of tilt-excess can be used to compute the mean

~curvature vectorH in terms of the local geometry ofμ which had already beenμ

notedin[Sch01,Lemma6.3]. In[Sch04b]Sch¨atzleprovidestheabovementioned

2link to C rectiﬁability as follows:

Theorem 3.1 in [Sch04b]. If μ is an integral n varifold in U satisfying (H )2

then the following two statements are equivalent:

2(1) μ is C rectiﬁable.

(2) For μ almost every x∈U there holds

2 2

tiltex (x,%,T μ) =O (% ), heightex (x,%,T μ) =O (% ) as %↓ 0.μ x x x xμ

2The quadratic decay of heightex implies C rectiﬁability without the con-μ

dition (H ) as may be seen from the proof in [Sch04b]. However, (1) would not2

2nimply (2) ifμ were merely required to satisfy (H ) for somep with 1≤p< ,p n+2

an example will be provided in C.5. On the other hand, it is evident from the

Caccioppoli type inequality relating tiltex to heightex and mean curvature,μ μ

see e.g. [Bra78, 5.5], that quadratic decay of heightex implies quadratic decayμ

for tiltex under the condition (H ). This leads to the following question:μ 2

2Problem. Does quadratic decay of tiltex imply quadratic decay of heightexμ μ

under the condition (H )?2

More generally, suppose thatμ is an integraln varifold inU satisfying (H ),p

1≤p≤∞, and 0<α≤ 1, 1≤q<∞. Does

R 1/q−α−n/q qlimsupr |T μ−T μ| dμ(ξ) <∞ξ xB (x)rr↓0

for μ almost all x∈U imply

R 1/q−1−α−n/q q

limsupr dist(ξ−x,T μ) dμ(ξ) <∞xB (x)rr↓0

for μ almost all x∈U?

The answer to the second question will be shown in 2.8–2.10 to be in the

np

aﬃrmative if and only if either p ≥ n or p < n and αq ≤ , yielding inn−p

particular a positive answer to the ﬁrst question. The main task is to prove the

following theorem which in fact provides a quantitative estimate together with

qthe usual embedding in L spaces.

Theorem 2.8. Suppose Q ∈N, 0<α ≤ 1, 1 ≤p ≤n, and μ is an integral n

varifold in U satisfying (H ).p

Then the following two statements hold:

nq np1 1(1) If p<n, 1≤q <n, 1≤q ≤ min{ , · }, then for μ almost all1 2 n−q α n−p1

na∈U with θ (μ,a) =Q there holds

−α−1−n/q2 qlimsupr kdist(·−a,T μ)ka 2L (μxB (a))r

r↓0

−α−n/q1 q≤ Γ limsupr kT −T μkμ a 1(1) L (μxB (a))r

r↓0

where Γ is a positive, ﬁnite number depending only onm, n, Q, q , and(1) 1

q .2

n(2) If p =n, n<q≤∞, then for μ almost all a∈U with θ (μ,a) =Q there

holds

−α−1limsupr kdist(·−a,T μ)k ∞a L (μxB (a))r

r↓0

−α−n/q≤ Γ limsupr kT −T μk q(2) μ a L (μxB (a))r

r↓0

where Γ is a positive, ﬁnite number depending only on m, n, Q, and q.(2)

Here T denotes the function mapping x to T μ whenever the latter exists.μ x

The connection to higher order rectiﬁability is provided by the following simple

adaption of [Sch04b, Appendix A].

Lemma 3.1. Suppose 0 < α ≤ 1, μ is a rectiﬁable n varifold in U, and A

denotes the set of all x∈U such that T μ exists andx

R

−n−1−αlimsup% dist(ξ−x,T μ)dμ(ξ)<∞.xB (x)%%↓0

1,αThen μxA is C rectiﬁable.

3The analog of Theorem 2.8 in the case of weakly diﬀerentiable functions can

be proved simply by using the Sobolev Poincar´e inequality in conjunction with

an iteration procedure. In the present case, however, the curvature condition is

needed to exclude a behaviour like the one shown by the function f : R → R

deﬁned by

∞X

−if(x) = (2 )χ −i−1 −i (x) whenever x∈R[2 ,2 [

i=0

1at 0; in fact an example of this behaviour occurring on a set of positive L

1/2measure is provided byf ◦g whereg is the distance function from a compact

1set C such that L (C)> 0 and for some 0<λ< 1

−3/2 1liminfr L ([x+λr,x+r[∼C)> 0 whenever x∈C.

r↓0

Therefore the strategy to prove Theorem 2.8 is to provide a special Sobolev

Poincar´e type inequality for integral varifolds involving curvature, see 2.4. In

the construction weakly diﬀerentiable functions are replaced by Lipschitzian Q

m ∼valuedfunctions, aQvaluedfunctionbeingafunctionwithvaluesinQ (R ) =Q

m Q(R ) ∼ where ∼ is induced by the action of the group of permutations of

m Q{1,...,Q} on (R ) .

Roughly speaking, the construction performed in a ballB (a)⊂U proceedsr

as follows. Firstly, a graphical part G of μ in B (a) is singled out. The com-r

plement of G can be controlled in mass by the curvature, whereas its geometry

cannot be controlled in a suitable way as may be seen from the example in C.2

used to demonstrate the sharpness of the curvature condition. On the graphical

part G the varifold μ might not quite correspond to the graph of a Q valued

functionbutstillhave“holes”or“missinglayers”. Nevertheless,itwillbeshown

that μ behaves just enough like a Q valued function to make it possible to re-

duce the problem to this case. Finally, for Q valued functions Almgren’s bi

m mPLipschitzian equivalence of Q (R ) to a subset ofR for some P ∈N whichQ

is a Lipschitz retract of the whole space directly yields a Poincar´e inequality.

More details about the technical diﬃculties occurring in the construction and

how they are solved will be given at the beginning of Section 1.

To conclude the introduction, it will be indicated why tilt quantities with

exponent diﬀerent from 2 may become relevant. The above mentioned decay

rates for tilt-excess (or height-excess) shown by Brakke in case the integral

1,1/2varifold μ satisﬁes (H ) imply that μ is C rectiﬁable but for every 1/2 <1

1,αα ≤ 1 there is no example known to the author of such a μ which is not C

1rectiﬁable. In contrast, for any 1/2+ < α ≤ 1, n > 1, there exists an

2(n−1)

example, see C.4, showing that tilt-excess and height-excess do not decay with

1,αpower 2α, i.e. the power corresponding to C rectiﬁability via Theorem 2.8

and Lemma 3.1. The 1 tilt does behave better in this respect. In fact, it will

2be shown that decay of the 1 tilt implies C rectiﬁability and locality of mean

curvature:

Lemma 3.2. Suppose μ is an integral n varifold in U satisfying (H ) and A1

denotes the set of all x∈U such that T μ exists andx

R

−1−nlimsup% |T μ−T μ|dμ(ξ)<∞.ξ xB (x)%%↓0

42Then μxA is C rectiﬁable and for every n dimensional submanifold M of

n+m 2

R of class C there holds

~ ~H (x) =H (x) for μ almost every x∈A∩Mμ M

~ ~where H denotes the mean curvature of M and −H corresponds to the ab-M μ

solutely continuous part of δμ with respect to μ.

The ﬁrst part of the lemma is a direct consequence of Theorem 2.8 and

Lemma 3.1 whereas the second part is an adaption of [Sch01, Lemma 6.3] with

the help of the diﬀerentiation theorem provided in B.1.

Theworkisorganisedasfollows. InSection1theapproximationofμbyaQ

valued function is constructed. In Section 2 the approximation is used to prove

the Sobolev Poincar´e type inequality 2.6 and Theorem 2.8. Section 3 provides

Lemmas 3.1 and 3.2. The Appendices A and B provide more basic properties of

rectiﬁable varifolds which are needed to prove the various results contained in

the body of the text in a precise fashion which then allows the example given in

Appendix C to demonstrate the sharpness of these results. Finally, Appendix

D collects for the convenience of the reader the results needed from Almgren’s

Big Regularity Paper [Alm00].

The notation follows [Sim83]. Additionally to the symbols already deﬁned,

imf and dmnf denote the image and the domain of a function f respectively,

⊥T is the orthogonal complement ofT forT ∈G(n+m,n),γ denotes the bestn

constant in the Isoperimetric Inequality as deﬁned in A.3, andf(φ) denotes the

−1ordinarypushforwardofameasureφbyafunctionf,i.e. f(φ)(A) :=φ(f (A))

wheneverA⊂Y,ifφisameasureonX andf :X →Y. Deﬁnitionsaredenoted

by ‘=’ or, if clarity makes it desirable, by ‘:=’. To simplify veriﬁcation, in case a

statement asserts the existence of a constant, small (ε) or large (Γ), depending

on certain parameters this number will be referred to by using the number of

the statement as index and what is supposed to replace the parameters in the

order of their appearance given in brackets, for example ε (m,n,1−δ /2).A.10 3

Acknowledgements. TheauthoroﬀershisthankstoProfessorReinerSch¨atz-

le for guiding him during the preparation of this dissertation as well as inter-

esting discussions about various mathematical topics. The author would also

like to thank Professor Tom Ilmanen for his invitation to the ETH in Zu¨rich

in 2006, and for several interesting discussions concerning considerable parts of

this work.

1 Approximation of integral varifolds

n+mIn this section an approximation procedure for integral n varifolds μ in R

by Q valued functions is carried out. Similar constructions occur in [Alm00,

Chapter 3] and [Bra78, Chapter 5]. Basically, a part of μ which is suitably

nclose to a Q valued plane is approximated “above” a subset Y of R by a

Lipschitzian Q valued function. The sets where this approximation fails are

nestimated in terms of μ and L measure.

In order to obtain an approximation useful for proving the main lemma 2.4

for the Sobolev Poincar´e type inequalities 2.6 and 2.8 in the next section, the

following three problems had to be solved.

5Firstly,intheabovementionedestimateonecanonlyallowfortiltandmean

curvature terms and not for a height term as it is present in [Bra78, 5.4]. This

is done using a new version of Brakke’s multilayer monotonicity which allows

for variable oﬀsets, see 1.8.

Secondly, the seemingly most natural way to estimate the height ofμ above

the complement of Y, namely measure times maximal height h, would not pro-

duce sharp enough an estimate. In order to circumvent this diﬃculty, a “graph-

ical part” G of μ deﬁned mainly in terms of mean curvature is used which is

largerthanthepartwhereμequalsthe“graph”oftheQvaluedfunction. Points

inG still satisfy a one sided Lipschitz condition with respect to points aboveY,

see 1.10 and 1.14(4). Using this fact in conjunction with a covering argument,

n¯ ¯the actual error in estimating theq height in a ballB (ζ) where L (B (ζ)∩Y)t t

n n 1/q¯ ¯and L (B (ζ)∼Y) are comparable, can be estimated by L (B (ζ)∼Y) ·tt t

n 1/q¯instead of L (B (ζ)∼Y) ·h; the replacement of h by t being the decisivet

nq∗ ∗improvement which allows to estimate the q height (q = , 1 ≤ q < n)n−q

instead of the q height in 2.4.

Thirdly, to obtain a sharp result with respect to the assumptions on the

mean curvature, all curvature conditions are phrased in terms of isoperimet-

ric ratios. Therefore, it seems to be impossible to derive lower bounds for the

density and monotonicity results for the density ratios by integration from the

monotonicity formula, see e.g. [Sim83, (17.3)], as in [Sim83, Theorems 17.6,

17.7]. Instead, lower bounds are obtained via slicing from the Isoperimetric

Inequality of Michael and Simon, see Appendix A, and it is shown that noninte-

gral bounds for density ratios are preserved provided the varifold is additionally

close to aQ valued plane, see 1.4. Both results appear to be generally useful in

deriving sharp estimates involving mean curvature.

n+m1.1. Ifm,n∈N,a∈R , 0<r<∞,T ∈G(n+m,n), andμ is a stationary,

integral n varifold in B (a) with T μ = T for μ almost all x ∈ B (a), thenxr r

⊥ ⊥T (sptμ) is discrete and closed in T (B (a)) and for every x∈ sptμr

n ny∈B (a), y−x∈T implies θ (μ,y) =θ (μ,x)∈N;r

hence with S ={y∈B (a):y−x∈T}x r

n nμxS =θ (μ,x)H xS whenever x∈B (a).x x r

A similar assertion may be found in [Alm00, 3.6] and is used in [Bra78, 5.3(16)].

1.2 Lemma. Suppose 0<M < ∞, M ∈/ N, 0<λ <λ < 1, m,n ∈N, T ∈1 2

n+mG(n+m,n), F is the family of all stationary, integral n varifolds in B (0)1

such that

n+m n+mT μ =T for μ almost all x∈B (0), μ(B (0))≤Mω ,x n1 1

and N is the supremum of all numbers

n −1 n+m¯(ω r ) μ(B (0))n r

corresponding to all μ∈F and λ ≤r≤λ .1 2

Then for some μ∈F and some λ ≤r≤λ1 2

n −1 n+m¯N = (ω r ) μ(B (0))<M.n r

6Proof. The proof uses the structure of the elements ofF described in 1.1. Since

n −1 n+m¯(ω r ) μ(B (0))n r

depends continuously on (μ,r)∈F ×[λ ,λ ], the ﬁrst part of the conclusion is1 2

a consequence of the fact that F is compact with respect to the weak topology

(cf. [All72, 6.4]). To prove the second part, one notes

2 2 n/2 2 n/2 n

(r −% ) < (1−% ) r whenever 0<%≤r< 1

and computes

X

n+m n 2 ⊥ 2 n/2¯μ(B (0)) = θ (μ,x)ω (r −|T (x)| )nr

n+m¯ ⊥x∈B (0)∩T (sptμ)r

X

n ⊥ 2 n/2 n≤ θ (μ,x)ω (1−|T (x)| ) rn

n+m¯ ⊥x∈B (0)∩T (sptμ)r

n+m n n≤μ(B (0))r ≤Mω r .n1

If sptμ 6⊂ T, then the ﬁrst inequality in the computation is strict. Otherwise,

the last inequality is strict because M ∈/N.

1.3 Remark. Any μ∈F satisﬁes

n+mn −1 n+m −1¯(ω r ) μ(B (0))→ω μ(B (0)) as r↑ 1,n r n 1

and this number may equal M. Therefore the conclusion N <M would fail if

λ = 1. However, the supremum in the deﬁnition ofN can be extended over all2

0<r≤λ r with N <M still being valid as will be shown in 1.4.2

1.4 Lemma (Quasi monotonicity). Suppose 0 < M < ∞, M ∈/ N, 0 < λ < 1,

and m,n∈N.

Then there exists a positive, ﬁnite number ε with the following property.

n+mIf a ∈R , 0 < r < ∞, μ is an integral n varifold in B (a) with locallyr

bounded ﬁrst variation,

n

μ(B (a))≤Mω r ,nr

and whenever 0<%<r

1−1/n¯ ¯kδμk(B (a))≤εμ(B (a)) ,% %

R

¯|T μ−T|dμ(x)≤εμ(B (a)) for some T ∈G(n+m,n),¯ x %B (a)%

0(here 0 := 1), then

n¯μ(B (a))≤Mω % whenever 0<%≤λr.n%

Proof. Using induction, one veriﬁes that it is enough to prove the statement

2with λ r ≤ % ≤ λr replacing 0 < % ≤ λr in the last line which is readily

accomplished by a contradiction argument using 1.2 and Allard’s compactness

theorem for integral varifolds [All72, 6.4].

7