Площадь образов измеримых множеств на многообразиях Карно глубины два с сублоренцевой структурой

• 1.    Preliminaries

The paper continues the research of metric properties of Lipschitz images on sub-Lorentzian structures started in [1]. Recall that these structures are a nonholonomic generalization of Minkowski geometry (see [2] and references therein). Research into both the structures themselves and their applications in physics [3, 4] began relatively recently. Article [5] is one of the first works in which such structures were studied. For further acquaintance with recent results established for sub-Lorentzian structures and their generalizations (for example, in the case of a multidimensional timelike coordinate), see, e. g., [6] and the list of cited literature.

• ^# The research is carried out in the framework of Russian state assignment for the Sobolev Institute of Mathematics, project № FWNF-2022-0006.

(0 2024 Karmanova, M. B.

In the current paper, we consider a mapping of Carnot manifolds. Despite the fact that locally they are constructed close to Carnot groups, these structures are of more general nature. Moreover, we weaken the requirements on a domain of a mapping: it can be defined on a measurable set which is not definitely compact. This weakened requirement influences the definition of sub-Lorentzian Hausdorff measure.

We consider the model case when the depth of domain and target space equals two.

Definition 1 (see, e. g., [7], cp. [8-10]). Fix a connected Riemannian C “ -manifold M of the topological dimension N . M is called a Carnot manifold of the depth two, if its tangent bundle T M contains a subbundle H = H M ^ T M of the dimension dim H independent of points from M such that for each point p G M there exists a neighbourhood U C M with a collection of C ⁱ -smooth basis fields X i , ... ,X n possessing the following property: at each point v G U we have T M = span { H, [H, H ] } .

The subbundle H M is called horizontal. If a basis vector field belongs to H M then its degree deg X k equals one, and it is called horizontal. The degree of vector fields different from horizontal equals two.

We may assume that the basis vector fields are C ⁱ -smooth.

Put n i = dim H and П 2 = N — dim H .

Let us describe the local homogeneous group and its basic properties. It follows from Definition 1 that there exist functions { c ijk (v) } i,j,k such that

N

[X i ,X j ](v) = ^c ijk (v)X k (v), v G U, ij = 1,...,N.

k =i

Theorem 1 [9]. Fix u G M. The collection cij k (u), cijk = [0,

if deg Xi + deg Xj = deg Xk, else, defines the structure of nilpotent graded Lie algebra.

Construct Lie algebra g ^u with structure constants from Theorem 1 as nilpotent graded Lie algebra of vector fields { (X iu ) ' } i =i on R N such that the exponential mapping

R N Э (x _i , ..., x n ) H exp

ft ^•X0')

equals identity [11, 12]. Thus (see, e. g., [9, 13]), we infer (X U ) ' (0) = e i , i = 1,... ,N. Next, we use the exponential mapping

O u :

R N Э (x _i ,..., x n ) H exp

(ft и

( u )

to push-forward the fields { (X iu ) ' } ^N- i to the neighbourhood of u on M. This neighbourhood together with the vector fields X iu = (O u ) * (X iu ) ' and a group operation defined by them constitutes the structure of a local Carnot group and is denoted by G u M (see details in [9]).

Theorem 2 (see [8, 9, 13] for various cases) . If the basis vector fields on M are C ^smooth

-^^

then X _i ^u depend continuously on u , i = 1 , . . . , N .

Definition 2 (see, e. g., [14]). A two-step nilpotent graded (Lie) group is a connected simply-connected Lie group G such that its Lie algebra V is graded, i. e., is representable in the form

V = V 1 Ф V 2 ,   [V 1 ,V 1 ] c V 2 ,   [V 1 ,V 2 ] = [V 2 ,V 2 ] = { 0 } .

If we put [V i , V i ] = V 2 instead of [V i , V i ] C V 2 then G is called a Carnot group.

Definition 3. Let M be a Carnot manifold, and w = exp (^Ni WiXi)(v). Put d2(v, w) = max *

( E j 2 ■ ( E j ²²

j : deg X j =1            j : deg X j =2

The set { w G M : d 2 (v,w) <  r } is called a ball with respect to d 2 of the radius r >  0 centered at v, and is denoted by B0X 2 (v,r).

It is easy to see that the Hausdorff dimension of M with respect to d 2 equals v = n i + 2n 2 .

Definition 4. Define a set function H ^v for A C M as

H ^v (A) = ш _{П 1} ^

• lim inf δ → 0

( E ^U i ∈ N i ∈ N

Box 2 (y i ,r i ) D A, y i G A, r i <5

where the infimum is taken over all coverings of A .

To this end, the symbol ω l stands for a volume of a Euclidean ball of the unit radius in R ^l .

The set function H ^ν is a measure (i. e., H ^ν is countably additive on Borel sets); it follows from the quasi-additivity of H ^ν and results of [15, 16]. It is also easy to prove that H ^ν and H ^N are absolutely continuous one with respect to another, and are doubling. The derivative of H ^N with respect to H ^V at x G M equals ^det(g(x)), where g is a Riemann tensor on M.

Consider one more Carnot manifold M with horizontal subbundle H and basis vector fields X i , ... ,X n . Denote the quasimetric constructed on M in the same way as in Definition 2 (with obvious changes) by d 2 . Put n i = dim H and П 2 = N — dim H .

Let us pass to the sub-Riemannian analogue of differentiability for our case, and to some important results.

Definition 5 [17]. Let M and M be Carnot manifolds, Q C M, and p : Q G IM. The mapping ϕ is hc-differentiable, or differentiable in the sub-Riemannian sense, at the (limit) point x G Q if there exists a horizontal homomorphism Lx : GxM G G^(x)M of local Carnot groups such that d2(p(y), Lx(y)) = o(1) • d2(x,y), where o(1) G 0 if Q Э y G x.          (1)

The hc-differential (or sub-Riemannian differential) L x is denoted by Dp(x).

Remark 1. By Local Approximation Theorem [8, 9, 13], we may consider in Definition 5 the quasimetrics defined like d 2 and d 2 on local Carnot groups instead of initial ones.

Definition 6. Let M and M be Carnot manifolds, Q C M, and p : Q G IM .If p is a Lipschitz mapping with respect to quasimetrics d2 and d2 then ϕ is Lipschitz in the intrinsic sense.

The next result was obtained for the first time by S. K. Vodopyanov [17].

Theorem 3. Let M and M be Carnot manifolds, D C M be a measurable set, and p : D G M be a Lipschitz mapping in the intrinsic sense. Then it is hc -differentiable almost everywhere. Moreover, the matrix of Dip consists of the “diagonal” (dim V k x dim V k ) -blocks, while all other elements vanish.

To this end, by “diagonal” ones, we mean the blocks consisting of elements such that their lines’ numbers correspond to the (basis) fields of degree k from M, and columns’ numbers, to the fields of the same degree k from M, k = 1, 2. Denote these “diagonal” blocks constituting the matrix of Dip, by D k = D k p, k = 1, 2.

In the paper, we assume that D C M is measurable, the mapping p is Lipschitz in the intrinsic sense and continuously hc -differentiable in the topology of its domain, the set p - ¹ (p(D o )) is compact, where D q = { y £ D : rank Dp(y) < N } , and p is bijective on its image outside D q . Moreover, n i > n ^ and П 2 ^ П 2 , or П ^ ^п 1 and П 2 > П 2 . For each k = 1, 2, choose integers n - £ [0, n k — n k ]. Set n o = 0.

Definition 7. Let w = exp (E N Ii W i X i )(v). Put d 2 (v,w) to be equal to

1^

max k =1 , 2

Пк-1+Пк wj2

j = n k -i + n - + 1

—

nk-1+nk wj2

1       / n k -1 + n _k

• sgn I E w ²

' j = n k -i + n - +1

—

nk-1+nk    \ wj2

j = n k -1 +1   /

The set { w £ M : d 2 (v,w) < is denoted by Box _d (v,r).

r ² } is called the ball in d 2 of the radius

r

> 0 centered at v and

To study the metric properties of surfaces lying in M, it is enough to consider the above analogue of the squared distance d ² ₂ , without considering the roots of the quantities involved in the definition. As in [1], we are going to study spacelike surfaces whose intersections with d ² ₂ -balls are bounded if the centers of these balls lie on this surface. Let us formulate the requirements on ϕ providing the spacelikeness of its image (see [1] and [6] for details).

----

.—-

Assumption 1. In each “diagonal” block D k of the sub-Riemannian differential D ϕ , denote the part consisting of first n_k lines, by D_k , and the rest, by D _k , k = 1,..., M. Consider such orthogonal mapping O k that transfers lines of D k to R ^r k x 0 n ^{k -r} k , where r k denotes the rank of D + , k = 1, 2.

To this end, assume that the rank of Dk equals the rank of Dk, k = 1,..., M. Moreover, for rk independent lines of DkOk with numbers i1 ,...,irk constituting the matrix L+k, the following holds: lengths of columns of [D-- Ok] (—+k)

¹ do not exceed 1/r k — c, c >  0

(see more details in [1]). Here [Q] denotes a part of a matrix Q consisting of its first r k columns, k = 1, 2. Note [1, 6] that this assumption provides the spacelikeness of the image of every mapping w H Dp(y) ( w ) , y £ D, in the sense of local Carnot groups G ^y M and G ^{p ( y )} M.

To describe the measure on the images of measurable sets, we will base on ideas of [1] to parameterize the fragment of intersection of the image of the measurable set with a d22-ball by a subset of some open connected set. Fix y £ D, e > 0, then [17] there exists such by(e) > 0 that if y,v £ D, and d2(y,w) < by(e) then max {d2 (Dp(y)(w),p(w)), d^y ^Dy^yy)wv)ppwv)) } < ed2(y,w).

Emphasize that here b y ( e ) depends on y. Moreover, in contrast to [1], to this end, the uniformity of b y ( e ) will not be required. Let us now formulate the new version of the definition of H _d ^ν -measure for images of measurable sets, and briefly describe the related terms.

Definition 8. Assume that A C M is a subset of the image of a measurable set D C M under Lipschitz in the intrinsic sense mapping ϕ which is hc-differentiable everywhere on its domain. Suppose that on D \ Dq, the mapping p is bijective on its image. Define Hdv(A) as

Шщ ш _{П 2} • liminf < V^ i r^v £ > 0     ²—'

i ∈ N

J ( Box^(xi,ri)) D A, Xi G A, ri < min{E, rxie}, i∈N bi = E if p 1(Xi) П Dq = 0,

and b i = 1 if X i G ^ ( D q )

,

where the infimum is taken over all coverings Ui e N (Box ^ (x i ,r i )) D A of A.

The main idea is also to parameterize the image of p by subsets of images of D p by means of the mapping n _x : ^(w) H Dp(y) ( w ) , x = p(y). Since locally G ^x M coincides with a neighborhood of x in M as set of points, we may consider π_x as a mapping of this neighborhood of x in M to itself. Namely, Box ^ (x,r) equals

n x ¹ ( Box _d (x, r) 0 ^x Im D p(y) ) ,                      if rank D p(y) = N and x / p ( D q ),

< U   n ,T, J _je ( Box _d (x,min { r,r _y,_£ } ) n x Im D p(y) ) , if rank D p(y) and x/p(D \ D q ),

y : ^ ( y )= x

'A ,                                                   if x G p(D \ D _q ) П p(D ₀ ),

where π _r,y,ε is the mapping

I

p ^{( w )} H D^y^w), p ^{( w ) H} Dp(y) ( w),

if w G Box 2 yy,r_p, y,_£) , p = min { r,r y,_£ } , and Dp(y) = 0, if w G Box 2 yy,r_r,_y,_£) , and Dp(y) = 0,

and A equals the union of assigned to x , and

П х¹ (Box d (x,r) n x Im Dp(y x )), where y x / D q , p(y _x ) = x, is

u

yED o : ^ ( y )= x

П г 1 _£ ( Box d (x, min { r, r y^} ) n x Im D p(y) ) .

The radii r _x , _£ , r y , _£ , and r s , y , _£ are defined the way similar to [1] with the replacement of 5 ( e ) by b y ( e ) . Roughly speaking, these values provide the “uniformity” of o(1) from (1) for measurable (not necessarily compact) sets by shrinking the balls in the preimage; see detailed definitions in [1].

The main result for this set function H _d ^ν is

Theorem 4. The function M D A H H ^ ^v (p(A)) is differentiable almost everywhere with respect to the measure H ^v : for almost all y G D the limit

li_m H d ^{v (} P ^(Box 2 ⁽ y,^{r )))} r > 0 H ^v (Box ₂ (y,r))

= D h v H (y)

exists. Moreover, this function is recoverable by its derivative: for A C D , we have

H (p(A)) = J D h v H (y) dH ^v (y).

A

The main steps of proving Theorem 4 and, in particular, quasi-additivity of the set function are the following results of independent interest.

Lemma 1. If В C D is such that p ¹ (p(D o j И B j is closed, where B = p^j , then Theorem 4 from [1] holds for В .

In contrast to [1], the constraint in Lemma 1 is related more to D q than to В since the latter need not to be a closed set.

Lemma 2. Let В be a ball in M ( with respect to the distance у/ (- ,g-) j . Then for each 5 >  0 , there exists an embedded closed ball 13 C В such that

H^v^В) )- H&(В )) | <5.

Emphasize that Definition 8 does not imply that H _d ^ν is a measure. For this reason, the assertion of Lemma 2 is not obvious as it can seem to be. Moreover, it is also not applicable to use the exhaustion of an arbitrary measurable set by compact ones by the same reason.

The main application of Theorem 4 is the area formula.

Theorem 5. Let M and M be Carnot manifolds, D C M be a measurable set, and the intrinsically Lipschitz mapping p : D ^ M be continuously hc -differentiable in the topology of its domain. Assume that p is bijective on its image on D \ D q , and Dp enjoys conditions of Assumption 1 everywhere. Then

7det (D + p(yj * D + p(yj - D — p(yj*D - p(y)           (2)

k =1

for almost all y E D \ p ^{- 1} (p(D o jj , and the area formula

Пу ^det ( D + p ⁽ y ^j * ^D + p ⁽ y ^{j - D} _kp ⁽ y ^j * D _kp ⁽ y ^j ) dH ⁽ yj = H ⁽ p ^{( A jj}

A A D ^{k 1}

holds for each measurable A C D .

The main idea of the proof of Theorem 4 is to calculate the derivative D h v H ^ ^v (yj explicitely at density points y E D of D. Fix y E D and consider В = Box 2 (y,rj and 13 = Box ₂ (y, 3 j, 3 > r, such that Box ₂ (y, 3 j 0 p ^{- 1} (p(D o jj = 0, and the covering { Box * (x i , r i j } i e N of the image p(B j from Definition 8. Put X i = p(y i j, i E N. From [5] it follows that

2       ,-------------------------------------------------------------------------------------------------

V ^det (^D k⁺ P(У i ⁾ * ^D fc P(У i ^{j - D}k p ⁽ y i j ^Dk p ⁽ y i j) • H y i (Dp ⁽ y i ^{j 1 (Box} d ⁽ x i ,r i ^jj) k =1

up to the multiple 1 + o(1j, where H y ^v is a Hausdorff measure constructed in the local Carnot group G ^y i M. Moreover, we may assume without loss of generality that e >  0 from Definition 8 is such that Dp(y _i j ^{- 1} (Box _d (x _i ,r _i jj C В, i E N. In turn, the right-hand side equals

• ² ,--------------------------------------------------------------------------------------------------

• V det (D+p(yi j*Dkp(yij - Dkp(yi j*Dkp(yij) • H (Dp(yij 1(Boxd(xi,rijj) k=1

up to the multiple 1 + o(1j, i E N. Here o(1j is uniform due to continuity of Riemann tensors on local Carnot groups and to Theorem 2. Definition 8 infers

Dp(yi j-1( Boxd(xi,rij) П D = p-1 (Box*1 (xi,rij), thus, the collection {D^(yi) 1(Boxd(xi,r))}ieN covers B И D. Since y is a density point of D then given a > 0, there exists ro > 0 such that if r < ro then

H ^v (B П D) H ^v (B )

H ^v (Box 2 (y,r) П D) > 1

— a.

H ^v (Box 2 (y,r))

It is left to show that if r < ro where ro > 0 is chosen for arbitrarily small a > 0, then inf ] E HV D^(yi) 1(BoXd(xi, ri))) : Xi G ^(B) > — Hv(B) i∈N

does not exceed a • H ^v (B). Assume that r and r are such that H ^v (B \ B ) < (a/2) • H ^v (B ), and choose the appropriate e >  0 from Definition 8 to provide the inclusions D ^(y _i ) ^(Boxd^ ,r _i )) C B, i G N. To prove the estimate (3) it suffices

• •    to apply Vitali’s Theorem for obtaining the collection {D ^(y _i ) ^{- 1} (Box _d (x _i , r _i )) } _{i e} N that covers B И D up to a set of H ^v -measure zero;

• •    for this set of H ^v -measure zero, to choose a covering by balls { Box 2 (y j ,r j ) } j e N such that the sum of their H ^v -measures does not exceed given e > 0;

• •    to replace { Box ₂ (y j ,r j ) } j _eN by {D^(y j ) ^{- 1} (Box _d (x j ,Kr j )) } j _eN such that

D^(y j ) (Box d (x j ,Kr j )) D Box 2 (y j ,r j ),

K .--------- K

= max { 1, Lip(^) } <  от , j G N; then the sum of their measures would not exceed Ke, <  от . The value e >  0 can be chosen in a such a way that Ke < (a/2) • H ^v (B ).

From here it follows that

(1 — a)Hv(B) < Hv(B И D) < £Hv (D^(yi)-1(Boxd(xi, ri))) + Ke i∈N

< Hv(B) + 2 • 2 Hv(B) = (1 + a)Hv(B), thus, the equality (2) and the area formula are true. The theorem follows.

Remark 2. All proofs do not require the uniformity of o(1) from the definition of hc-dif-ferentiability.