Approximation of Submanifolds

11. Approximation of Submanifolds

is a barycentric mapping with respect to the induced metric g|S, theny is called an intrinsic Karcher triangulation. If moreover each y|re is also a barycentric mapping with respect to the metric ofM, thenyis called an extrinsic Karcher triangulation.

Goal. The possibility for the existence of an intrinsic Karcher triangulation has been dealt with in section 8. The question of this section will now be how well an extrin

sic Karcher triangulation, induced by the same complex K and the same vertex set {pi}, approximatesS. Note that such an extrinsic Karcher triangulation is always an interpolation of the given triangulation ofS in the sense of 7.14.

11.2 Proposition. Let y be a piecewise (ϑ, h)-small intrinsic Karcher triangulation y of S with ||Wν|| ≤κ for all Weingarten maps Wν. Suppose that all vertices pi =y(ri), i ∈e, lie in a common convex ball with respect tog for eache∈ Kⁿ. Then for small edge lengths `ij := dS(pi, pj) and `¯ij :=d(pi, pj) with respect to g|S and g, it holds

|`ij−`¯ij|.κhϑ⁻¹`¯ij.

Proof. There exists an extrinsic Karcher triangulationxof some set S⁰⊂M with the same combinatorics and vertices asy(that means: interpolatingy) because the vertices of each simplex are contained in a convex ball. We do not know if S⁰ is a manifold, because the fullness of the extrinsic simplices is not clear a priori, but will be a result of the length estimate.

Let us show the claim for the edge γ : e_i ; e_j in re with tangent v := ˙γ. The estimate 7.9 can be extended to the whole edge:

|(dx−P dy)v|.hϑ⁻¹|(∇dx−P∇dy)(v, v)|

|dy v|

As edges are mapped to geodesics, ∇dx(v, v) = 0 and

t

∇dy(v, v) = 0. And as h∇dy vdy v, νi=−h∇dy vν, dy vifor any normalν toS, we have|∇dy(v, v)| ≤κ|dy v|². So

|`<sub>ij</sub>−`¯_ij| ≤ ˆ

|dx v| − |dy v|

≤ ˆ

|(dx−P dy)v| ≤κhϑ⁻¹ ˆ

|dy v|,

q. e. d.

11.3 Corollary. Situation as before, additionally||Wν||+h||∇Wν|| ≤κfor all Weingarten maps. Let`¯ij also induce a(ϑ, h)-full metricg^e onrK. Then for smallh,

|(y^∗g−g^e)hv, wi|.(C0h²+κhϑ⁻¹)|v| |w|, (11.3a)

|∇^y_v^∗gw− ∇^g_v^ew|.C˜_0,1⁰ h|v| |w|, (11.3b) where C˜0,1 = (C0,1+κ²)ϑ⁻¹. The second estimate also holds for any other piecewise at metric g^eon rK.

Proof. The metric estimate comes from the edge length comparison above, and the connection estimate from 6.23 does not depend on the chosen metric, as long as it is at. Due to the Gauÿ equation (e. g. Jost 2011, thm. 4.7.2), the intrinsic curvature tensor ofS is bounded byC0+||Wν||²and its derivative byC1+||Wν||||∇Wν||, q. e. d.

11. Approximation of Submanifolds

11.4 Remark. The observation that 11.3b also holds for any other piecewise at metric on Kmeans that ifgis approximated up to second order by a better-suited approximation of edge lengths `<sub>ij</sub> than just `¯_ij, then the approximation of the connection remains unchanged.

Nevertheless, taken as it is, 11.3 says that a simple interpolation of a given triangula

tion, just as in Euclidean space, is not the best candidate for geometry approximation.

Henceforth, the rest of this section is devoted to the normal graph mapping, which reveals better approximation properties.

General Properties of Normal Graphs

11.5 Denition. LetS ⊂M be ann-dimensional compact boundaryless smooth subman

ifold. A second submanifoldS⁰ ⊂M is said to be a normal graph overS if there is a normal vector eldZ onS such that

Φ :a7→exp_aZ|_a (11.5a)

is a bijective mapping S → S⁰. Where we need it, we will also consider the smooth transitionS;S⁰ via the homotopy

Φt:a7→exp_atZ|a. (11.5b)

Parallel transport alongt7→Φt(p)fromΦa(p)toΦb(p)will be denoted byP^b,a.

11.6 Remark. <a> The term normal graph or normal height map is mostly used in the context of triangular approximation of surfaces in R³, e. g. in Hildebrandt et al. (2006). In the context of manifold-valued pde's, it is more common to consider the geodesic homotopy Φt, see 13.6d, which also Grohs et al. (2013) use. In particular, their control of distortion alongΦt is equivalent to our control of the distortion byΦ.

<b> Here, as usual, we do not want to treat global properties ofM, so we always tacitly assume|Z|<injM.

<c> Any other n-dimensional submanifold S⁰ ⊂ M that is near enough to have a bijective orthogonal projectionS⁰ →S can be represented as a normal graph overS. Here normal projection means mapping somep∈S⁰ onto the pointq∈Sminimising d(p, q). The largest εsuch that the orthogonal projection Bε(S)→S is well-dened is called the reach ofS(introduced by Federer 1959, def. 4.1, for a recent overview see Thäle 2008). Another formulation for the same thing is that

Φ :˜ T S^⊥→M,(p, Z)→exp_pZ

is a dieomorphism fromOε:={ν∈T S^⊥:|ν| < ε} onto its image.

<d> It is well-known (see e. g. Hildebrandt 2012, eqn 1.11) that for M =R^m, the mapΦis locally a dieomorphism if |Z|||Wν||<1for all Weingarten mapsWν. (Note that this bound can only capture the local geometry ofSbut cannot see if some part of Sthat is intrinsically far from a pointp∈Scomes close topin the surrounding space M.) Dierent to the usual argument involing the curvature radius ofSand osculating

spheres, one can use Jacobi elds as in 1.25 (we use the notation from there) to see this:

We already knowdΦ_tp˙=J(t)for a Jacobi eld withJ(0) = ˙pandJ˙(0) =∇_p˙ν, so

d

dtΦ^∗_tghp,˙ pi˙ = ghJ,Ji˙ . Now let ν be a unit normal eld. Then J˙ =WνJ. If p˙ is the eigenvector in direction of the largest eigenvalue κ, we have

d

dtΦ^∗_tghp,˙ pi˙ = 2κ(t)|J(t)|,

whereκ(t)is the eigenvalue of the Weingarten mapWν in directionp˙atΦt(p). In our case R = 0, the Riccati equation 1.25b gives W˙ν =−W_ν², so the eigenvalues κi also evolve by κ˙i = −κ²_i. This dierential equation has solution κi(t) = (t− _κ¹

i(0))⁻¹ =

κ_i(0)

κ_i(0)t−1, hence

d

dtΦ^∗_tghp,˙ pi˙ = 2κ

κt−1Φ^∗_tghp,˙ pi.˙

This is solved byΦ^∗_tghp,˙ pi˙ = (1−κt)²ghp,˙ pi˙ , sog is positive denite forκt <1.

<e> If we represent Z = fⁱνi with parallel unit normal vector elds νi and scalar functions fⁱ, then ∇Z = dfⁱ⊗νi +fⁱ∇νi, which splits into tangential and normal parts

t

^{∇Z =fi∇ν}i,

n

^{∇Z=dfi⊗ν}i.

Hence, although ||∇Z|| usually shrinks slower than |Z| for |f| → 0, the tangential part is||

t

^∇Z||.κ|f|, whereκis an upper bound for the Weingarten mapsWν ofS. Similary,||

t

^∇2Z||.κ|df|+κ⁰|f| ifκ⁰ bounds all||∇Wν||.

11.7 Situation. Let S⁰ be given as a normal graph over S by a vector eld Z with d:=

|Z| ≤ε² and||dd|| ≤εeverywhere, and let the Weingarten maps ofS be bounded by

||Wν||+ε||∇Wν|| ≤κ. This means||∇Z||.κε. For simplicity, assumeκ≤κ².

11.8 Proposition. Situation as in 11.7. The mapΦ :a 7→exp_aZ is locally a dieomor

phism if|d|(κ+√

C₀)<1 everywhere.

Proof. Let us supposeZ has unit length at some point and see for whichtthe mapΦt

is locally a dieomorphism. Note that 1.25b gives|κ˙i| ≤C0+|κ²_i| for any eigenvalue of a Weingarten map. The equationu˙ =−C0−u²leads to a subsolution

u(t) =p

C₀tanp

C₀t−arctan^κ√i(0)

C0

.

Regarding _dt^dΦ^∗_tghp,˙ pi ≤˙ κ(t)Φ^∗_tghp,˙ pi˙ , which has positive solutions (for positive initial data) as long asκis bounded, it suces to know where the rst pole ofκ_i(t)can occur.

The rst pole ofu(which must also bound the position of the rst pole ofκ_i) is where

√C₀t−arctan^κ√i(0)

C0 =±^π₂. Now a simple function inspection shows 1

1 +s < ^π₂+ arctans, − 1

1 +s >−^π₂ + arctans, so there will be no pole as long as|√

C₀t|<(1 +^√κ

C₀)⁻¹, q. e. d.

11. Approximation of Submanifolds

11.9 Observation. By 1.16, the dierential ofΦ_tisdΦ_tV =J(t)for the Jacobi eld with

J(0) =V, J(0) =˙ ∇VZ. By 6.1, this means for Q_t:V 7→V +t∇VZ

|dΦ_tV −P^t,0Q_tV|.C₀d²t²(1 +κεt)|V| (11.9a) and

|dΦ_sV −P^t,0V|.κε|V|+C₀d²t²(1 +κεt)|V|. (11.9b) This estimate is scale-invariant with respect to scaling ofZ: IfZ⁰ =αZ, thenΦ⁰_t/α= Φt. SotZ=t⁰Z⁰ is scale-invariant, and the estimate only containstZ, neverZ alone.

But due to 11.6e, this distortion happens mostly in normal direction, the tangential change is of higher order:

||

t

^(dΦ−P)||.κd+C0d² (11.9c)

11.10 Proposition. Situation as in 11.7. Consider some point p∈reachS with projection

ψ(p) ontoS. Ifp= exp_ψ(p)dν for some unit normal vector ν with ||Wν|| ≤κ, and if κε < ¹₂, the orthogonal projectionψsatises

||dψ−Q⁻¹_ν

t

^Pp,ψ(p)||.C0d², ||dψ−

t

^Pp,ψ(p)||.κd+C0d², (11.10a) whereQν is the linear map TpS→TpM,V 7→V +d∇Vν. Ifdν is replaced by some other normal vector eldZ withexp_ψ(p)Z =p, andQ is replaced byV 7→V +∇VZ, it holds

||dψ−(

t

^Q)−1

t

^Pp,ψ(p)||.C0d², ||dψ−

t

^Pp,ψ(p)||.κd+C0d². (11.10b) Proof. Let us rst show that

t

^Qdoes not depend on howZis chosen at points6=ψ(p). IfZ =dν in a neighbourhood ofψ(p), whereν is a parallel unit normal eld anddis constant, thenZ is parallel, and so∇Z=

t

^{∇Z =dW}ν. For any otherZ,

t

^∇Z stays the same, and only some part

n

^{∇Z 6= 0}is added. That means

t

^Qν =Q_ν onT_ψ(p)S. So we will only prove 11.10b.

ad primum: Observe that the operator

t

^Q:TpS →TpS fullls ||

t

^Q−id||.κε < ¹₂ by assumption, hence is invertible with||(

t

^{Q)−1−id||} . _1−κε^κε by 6.15, which gives

||(

t

^Q)−1||.1 +_1−κε^κε = _1−κε¹ <2. Hence the claim is proven if we can show||

t

^(Qdψ−

P)||.C0d².

Consider some vectorV ∈TpM and splitV =Vp+Vν as in 1.25c. Then

t

^{Qdψ(V) =}

˙

p+

t

^∇p˙Z =J_p(0) + ˙J_p(0)on the one hand, and

t

^{P V =P V}p =P J_p(1)by 1.25d on the other. So 6.1 gives

|J_p(1)−P(J_p(0) + ˙J_p(0))|.C₀d²|p|˙ .C₀d²|V|.

ad sec.: We have||dψ−

t

^{P|| ≤ ||dψ−(}

t

^Q)−1

t

^P||+||(

t

^Q)−1

t

⁻

t

^||. The rst norm has been estimated above, and the second is.κdbecause||

t

^Q

t

⁻

t

^||.κddue to 11.6e

and the boundedness of(

t

^Q)−1, q. e. d.

Remark. ForC0= 0, this (exact) representation of the projection dierential is the one in Wardetzky (2006, thm. 3.2.1) and Morvan and Thibert (2004, lemma 4).

11.11 Proposition. Situation as in 11.7. Let

t

⁰ be the orthogonal projectionT M|S⁰ →T S⁰. Then for small ε, we have ||P

t

^{0 −}

t

^P|| . κε+C₀d². This means that the angles ](T_Φ(p)S⁰, P T_pS)and ]((T_Φ(p)S⁰)^⊥, P T_pS^⊥)between the corresponding tangent and normal spaces must be bounded by this factor, too. Therefore, normals ν_i toS can be extended to normal eldsν_i,t along Φ_t with|ν_i,t−P^t,0ν_i|.κε+C₀d².

Proof. For the time of this proof, let us write the terminal valueJ(1)of a Jacobi eld along t 7→ Φ_t(p) with initial values J(0) = ˙q and J˙(0) = ˙ν as T( ˙q,ν˙). Linearity of the Jacobi equation translates into linearity of T. In this notation, the splitting from 1.25c says that a vector V ∈T M|S⁰ can be represented asV =T( ˙p,

t

^{ν) +˙ T(0,}

n

^ν)˙ . We argue that its projection

t

^0V ontoT S⁰ is almostT( ˙p,∇p˙Z).

In fact, all tangent vectors on S⁰ have the formT( ˙q,∇q˙Z) for someq˙ ∈ T S. Now consider

|V −T( ˙q,∇q˙Z)|²=|T( ˙p−q,˙ ν˙− ∇q˙Z)|².

This is minimal among all r˙ if

t

^{0V =T( ˙q,∇}q˙Z), this means its norm has vanishing derivative in direction( ˙r,∇r˙Z). BecauseT is linear, this gives

0 =hT( ˙p−q,˙ ν˙− ∇q˙Z), T( ˙r,∇r˙Z)i for allr˙∈TpS.

Now recall thatT(U, W) =P(U+dW) +O(C₀d²), hence this is

=hp˙−q,˙ r˙+d∇r˙Zi+dhν˙− ∇q˙Z,ri˙ +d²hν˙− ∇q˙Z,∇r˙Zi+O(C0d²).

Ifp˙= ˙q, the rst term vanishes, and (using thatκdis small) the remaining ones are estimated from above by κd|V| |r|˙ . Because the minimisation is well-conditioned at this position, the optimalq˙ isp˙+O((κd+C0d²)|V|).

Now recall from 1.25d thatP

t

^{P V =T( ˙p,}

t

^∇p˙Z), which gives that the claim|P

t

^0V−

t

^{P V|=|(}

t

^0−P

t

^P)V|=|T(0,

n

^∇p˙Z)|+O((κd+C0d²)|V|).(κε+C0d²)|V|is just

the usual Jacobi eld estimate 6.1, q. e. d.

11.12 Corollary. Omitting the last paragraph of the proof, one gets||

t

^P

t

⁰⁻

t

^P||.κd+C0d². Remark. This is analogous to the classical statement ||P(Ph−1)P|| . dup to constants depending on the geometry from Dziuk et al., whereP is the projection ontoT SandPhthe projection ontoT S⁰.

Geometric Distortion by the Graph Mapping 11.13 Lemma. Situation as in 11.7. Then for Q:U 7→U+∇UZ,

|hQU, QVi − hU, Vi|.(κ²d+C₀d²)|U| |V| for allU, V ∈T_pS.

Proof. Just because hU +∇UZ, V +∇VZi − hU, Vi = h∇UZ, Vi+ h∇VZ, Ui +

h∇UZ,∇VZiand 11.9c, q. e. d.

11.14 Conclusion. Situation as in 11.7. Pulled back to S, the S⁰ metric Φ^∗g|phU, Vi = g|_Φ(p)hdΦU, dΦVifullls

(Φ^∗g−g)hU, Vi

.(κ²d+C₀d²)|U| |V|.

11. Approximation of Submanifolds Proof. Is a direct application of 11.9c and 11.13. We especially remark that the dier

ence between g|p andg|Φ(p) does not need to be handled explicitely, as P^p,Φ(p) is an

isometry with respect to these two metrics, q. e. d.

11.15 Remark. AsΦ^∗gandgare equivalent metrics,A:=dΦ^tdΦ(wheredΦ^tdenotes theg-adjoint

of dΦ) is a self-adjoint automorphism of TpS such thatΦ^∗ghU, Vi = ghAU, Vi, called the metric distortion tensor by Wardetzky (2006, p. 53). In the numerical literature, it is common not to compare the Riemannian metrics, but to estimate directly||^G_G^eA−id||, which already includes the volume element change (cf. the proof of 7.3), see Dziuk (1988); Demlow (2009); Heine (2005).

For a comparison with the tensorJ from 6.20, considerΨ := Φ⁻¹. IfM is the Euclidean space Rⁿand S⁰ is a piecewise at submanifold, its metric g^e :=g|S⁰ is piecewise at. The metricg|Spulls back to a metricΨ^∗gonS⁰, and there isJsuch thatΨ^∗ghU, Vi=g^ehJ U, Vi. So the transformationsAandJperform inverse tasks.

11.16 Proposition. Situation as in 11.7. For a given vector U and a vector eldV on S,

dene the connection distortion W := ∇dΦtUdΦtV −dΦt(

t

^∇UV). This vector eld obeys the dierential equation

W¨ =R(Z, W)Z+ ˙F forF :=R(Z, dΦtU)dΦtV +∇²_dΦtU,dΦtVZ with initial valuesW(0) = 0andW˙ (0) =F(0).

Proof. Let us abbreviateU^t:=dΦ_tU,V^t:=dΦ_tV, and denote the parallel translation ofZ along t7→exp_ptZ also asZ. LetK :=∇U^tV^t and J :=dΦ_t(

t

^∇UV). Then we want to determineW =K−J.

By 1.25,J is a Jacobi eld, i. e.J¨=R(Z, J)Z. An inhomogeneous Jacobi equation describesK:

K¨ =R(Z, K)Z+D_tR(Z, U^t)V^t+D_t∇²_Ut,V^tZ.

In fact, consider a variationγ(r, s)of geodesics (inM), i. e. we assume thats7→γ(r, s) is a geodesic for each xeds, with ∂_sγ(0,0) =V and ∂_rγ(0,0) =U. Transport this alongt asc(r, s, t) := exp_γ(r,s)tZ|_γ(r,s). Then we want to determineK=D_r∂_s, so we consider

K¨ =DtDtDr∂s=DtDrDt∂s+DtR(∂t, ∂r)∂s

=D_tD_rD_s∂_t+D_tR(∂_t, ∂_r)∂_s, the rst term of which is

∇Z∇U^t∇V^tZ =∇Z∇²_Ut,V^tZ+∇Z∇_∇

U tV^tZ

=∇Z∇²_Ut,V^tZ+∇∇_{U t}V^t∇ZZ+R(Z,∇U^tV^t), Z

=D_t∇²_Ut,V^tZ + 0 +R(Z, K)Z.

The initial value is computed in exactly the same way, q. e. d.

11.17 Proposition. Situation as before. IfC₀|Z|²+||∇Z|| ≤ ¹₂, then

|∇dΦUdΦV −dΦ

t

^∇UV|.|U||V|(||∇²Z||+C0|Z|) +C0|∇UV||Z|². If we only consider the tangential part

t

^W, then even

|

t

^∇dΦUdΦV −dΦ

t

^∇UV| .|U||V| ||

t

^{∇2Z||+||∇Z||+C}0|Z|²

+C0|∇UV||Z|². (11.17a) Proof. Preparatory step one: Let us rst establish the boundedness ofW =K−J and show|W|.|∇_UV|+at, wherea:=|U^t||V^t|(||∇²Z||+C₀|Z|): Thet-derivative ofK is, as ∂t=Z,

∇_Z∇_UtV^t=∇_Ut∇_ZV^t+R(Z, U^t)V^t

=∇U^t∇V^tZ+R(Z, U^t)V^t = ∇²_Ut,V^tZ+∇∇_{U t}V^tZ+R(Z, U^t)V^t,

so _dt^d|K| ≤ |K| ≤˙ a+||∇Z|| |K|. As|Z|is short by assumption, we have|U^t|.|U|and

|V^t|.|V|. The dierential inequality of the formu˙ ≤a+bugivesu≤(u0+^a_b)e^bt−^a_b. Forbt≤ ¹₂, this function is dominated byu0+ 2bt(u0+^a_b)≤2(u0+at).

For the bound on J, we have |J| ≤ |J(0)| +t|J˙(0)| as usual, and J(0) = ∇UV, J˙(0) =∇ZJ(0)shows that these terms are already contained in theK estimate.

Preparatory step two: Now let us show

W(t)−

ˆt

0

P^t,τF(τ) dτ

.C0t²|Z|²(|∇UV|+at).

The proof idea is from Jost (2011, thm 5.5.2). LetA:=´t

0P F. This is a vector eld fulllingA¨= ˙F with the same initial values asW, namelyA(0) = 0andA(0) =˙ F(0). Furthermore, let w: [0;t]→ R be the solution ofw¨ =C0|Z|²|W| with initial values w(0) = ˙w(0) = 0. Then, for some parallel vector eldE along t, dene v := (hW − A, Ei −w)/tand obtain

d

dt( ˙vt²) = d

dt (hW˙ −A, Ei −˙ w)t˙ − hW −A, Ei+w

= (hW¨ −A, Ei −¨ w)t¨ ≤0.

This means thatvt˙ ²≤0, hencev˙≤0. NowhW−A, Ei −vhas a double root att= 0, so v(0) = 0 and thus v ≤ 0 everywhere. And because E was arbitrary, this already means

|W −A| ≤w.

Therefore we are done if we can bound uby the right-hand side of the proposition.

But as we know that|W|.|∇UV|+at, we can simply integratew¨=C0|Z|²|W|twice and obtain the desired estimate (the argument is the same as in the proof of 6.5).

ad primum: The nal estimate for |W| comes from|´

P F| ≤ ´

|F|, together with

|F(t)|.|F(0)| ≤afort≤1, because the same holds forU^tandV^t, and the norm of

∇²Z|Φ_t(p) is the same as the norm of∇²Z|p becauseZ is parallel alongt7→Φ_t(p).

11. Approximation of Submanifolds ad sec.: For the estimate of|

t

^W|let

t

^tbe the orthogonal projection onto the tangent space ofΦ_t(S)and consider

t

^tW(t)−

ˆt

0

t

^{tPt,τF(τ) dτ =}

t

^tW(t)−

ˆt

0

P^t,0

t

^{tP0,τF(τ)+(}

t

^{tPt,τ−Pt,0}

t

^{P0,τ)F(τ) dτ.}

Then|

t

^{P F|} .|U||V|||

t

^∇2Z||, and the projection dierence is estimated by t||∇Z||+

C₀t²|Z|²in 11.11, q. e. d.

11.18 Theorem. Let y : rK → S be the triangulation of a smooth submanifold S ⊂ M

with Weingarten maps W_ν bounded by ||Wν||+h||∇Wν|| ≤ κ and x : rK → S⁰ an extrinsic Karcher triangulation with the same verticesp_iandy=ψ(x). Supposeg^eis a (ϑ, h)-small piecewise at metric onrKinduced by edge lengthsd_S⁰(p_i, p_j) =d(p_i, p_j). Then for smallh, it holdsd(x, y).h²ϑ⁻¹ ∇dx−P∇dy _L^∞ and

|(y^∗g−g^e)hv, wi|.(κ²d+C₀d²)|v||w|,

|∇^y_v^∗gw− ∇^g_v^ew|.κhϑ⁻² ∇dx−P∇dy L^∞|v||w|+ho.,

where ho. stands for higher-order terms whose coecients depend on C0, κ, ϑ,|v|,

|w| and|∇^g_v^ew|. The norm on the left-hand side may be induced by eitherx^∗g,y^∗g, or g^e, because all three are equivalent.

Proof. By the estimate 7.12 ford(x, y),S⁰ is a normal graph overS for small hwith

|Z| = d(x, y) . h²ϑ⁻¹ ∇dx−P∇dy . Morally, it is clear that ||∇Z|| must be con

trolled by ∇dx−P∇dy , too. In fact, we can precisely compute this forV =dy v:

∇VZ=∇dy v(−Xx|y) = ˙J(1)

for a Jacobi eld along x;y withJ(0) = dx v and J(1) =dy v by combining 1.23 and 12.3 (we postpone the computation to the next section because it is more relevant there), so by 6.1|∇VZ−(dx−P dy)v|.C0d²(x, y), hence 7.14 gives

||∇Z||.hϑ⁻¹ ∇dx−P∇dy +C0d². Now becausex= Φ◦y and hencedx=dΦdy, 11.14 gives

|(x^∗g−y^∗g)hv, wi|.(κ²d+C0d²)|v|y^∗g|w|y^∗g.

The comparison ofx^∗g andg^e is done in 6.17.Analogously, we only compare ∇^y∗g to∇^x∗g and refer to 6.23 for the rest: For a vectorv and a vector eldw, 11.17 gives (together with 11.6e)

|∇^S_{dx v}⁰ dx w−dΦ∇^S_{dy v}dy w|.κh²ϑ⁻¹ ∇dx−P∇dy +ho.

By denition of the pull-back connection,∇^S_{dx v}⁰ dx w=dx∇^x_v^∗gwand thus

|dΦdy(∇^x_v^∗gw− ∇^y_v^∗gw)|.κh²ϑ⁻¹ ∇dx−P∇dy +ho.

Together with||dΦ⁻¹||.1 and||dy⁻¹||.1/hϑ, this shows the claim, q. e. d.

The Weak Shape Operator

11.19 Lemma. Let S ⊂ M be a smooth submanifold with boundary ∂S in M. Let U be a smooth vector eld onS, not neccessarily tangential toS. ThenU may be extended to a vector eld onM in such a way that div^M

t

^{U = divS}

t

^U anddiv^M

n

^{U =−hU, Hi}, whereH=∇^M_eiei for any orthonormal basisei ofTpSis the mean curvature vector of S.

Proof. It suces to nd a local extension ofU to some small neighbourhood ofS. Let e1, . . . , en, νn+1, . . . , νmbe an orthonormal basis ofTpM. Thendiv^MU =h∇^M_eiU, eii+ h∇^M_νjU, νji. IfU is extended parallel in normal direction, the latter term vanishes.

Regarding the tangential part, observe h∇^M_ei

t

^{U, e}ii = h

t

^∇Me_i

t

^{U, e}ii, and because

t

^{∇M =∇S}, this is div^S

t

^U.

Now consider

n

^{U =αjν}j. Again, if U is constant in normal direction, div^M

n

^{U =}

h(∂iα^j)νj, eii+hα^j∇^M_eiνj, eii, the former term vanishes, the latter one isα^jh∇^M_eiνj, eii=

−α^jh∇^M_eiei, νji, q. e. d.

11.20 Lemma. Let S ⊂ M be a smooth submanifold with boundary ∂S in M. Then for smooth vector eldsV andW onM,

ˆ

S

hW, νidiv^MV +hW, νihV, Hi+h∇^M_V ν, Wi+hν,∇^M_V Wi= ˆ

∂S

hW, νihV, τi,

whereτ is the outer normal of ∂S inS.

Proof. Letf :=hW, νi. By the divergence theorem (Lee 2003, thm. 14.23), we have ˆ

S

div^S(f V) = ˆ

∂S

hf V, τi.

Now by product rule and 11.19,div^S(f V) =fdiv^SV +V f =fdiv^MV −fhV, Hi+

V f, q. e. d.

11.21 Corollary. Let S ⊂ M be a smooth submanifold without boundary. The operators sν, σν:X(T M|S)×X(T M|S)→R, dened by

sν(V, W) :=

ˆ

S

h∇^M_V ν, Wi+hW, νihV, Hi, σν(V, W) :=− ˆ

S

hW, νidiv^MV+hν,∇^M_V Wi

coincide for smoothSand each normal eldν. On tangential vector elds,s_ν(V, W) = σ_ν(V, W) =−´

II_ν(V, W). IfS were only piecewise smooth, σ_ν would still be well-de

ned. It is called the weak shape operator ofS.

11.22 Proposition. Situation as in 11.7. Then there are normal elds ν for S andν⁰ for S⁰ such thatσ_ν approximates the weak shape operatorσ_ν⁰0 of S⁰, which is

σ_ν⁰0(V, W) =− ˆ

S⁰

hW, ν⁰idiv^MV +hν⁰,∇^M_V Wi,

up to rst order: |(σν(V, W)−σ⁰_ν0(P V, P W)|.ε V _H1(T M|S) W _H1(T M|S).

12. Variation of Karcher Simplex Volume

Im Dokument numerical approximation in riemannian manifolds by karcher means (Seite 87-97)