[proofplan] We use Perelman's reduced-volume monotonicity in its equality-case form as a stronger rigidity theorem of which the present result is a corollary: the derivative of the reduced volume is an integral of a non-positive density, and vanishing of the derivative forces the reduced-distance tensor identity on the regular set away from the $\mathcal L$-cut locus. The cut locus has zero contribution to the reduced-volume measure, so the equality-case theorem identifies the regular generated region, namely the points reached by unique minimizing $\mathcal L$-geodesics before cut time, as the smooth region on which the tensor identity holds. That tensor identity is exactly the gradient shrinking Ricci soliton equation in backward time, and integrating the vector field $\nabla^{g(\tau)}l(\cdot,\tau)$ reconstructs the canonical soliton form. Conversely, for a canonical normalized shrinker, the reduced-distance density is time-independent after the canonical diffeomorphism and rescaling, so the reduced volume is constant. [/proofplan]

[step:Apply the equality case in reduced volume monotonicity]Let $n := \dim M$. Let $(p,0)$ denote the regular basepoint for the reduced distance, meaning that Perelman's reduced distance based at $(p,0)$ is defined and has the standard smoothness and cut-locus structure along minimizing $\mathcal L$-geodesics. Let $l: M \times I \to \mathbb{R}$ be that reduced distance. For each $\tau \in I$, define the reduced-volume density $v_\tau: M \to [0,\infty)$ by \begin{align*} v_\tau(x) = (4\pi\tau)^{-n/2} e^{-l(x,\tau)}. \end{align*} This density is taken with respect to the Riemannian volume measure $d\mu_{g(\tau)}$. Define the reduced volume $\widetilde V: I \to [0,\infty)$ by \begin{align*} \widetilde V(\tau) = \int_M v_\tau(x)\,d\mu_{g(\tau)}(x). \end{align*} We invoke Perelman's reduced-volume monotonicity and equality-case rigidity theorem as a packaged equality-case result; the present proof reconstructs the soliton form from that equality-case identity rather than reproving Perelman's monotonicity formula. In the form used here, for a complete backward Ricci flow with bounded curvature on compact time subintervals, reduced distance based at a regular point, and finite reduced volume, $\widetilde V$ is non-increasing. Moreover, if its [distributional derivative](/page/Distributional%20Derivative) vanishes on a compact subinterval, then the theorem gives all five conclusions needed below: the reduced-distance function is smooth on the pre-cut regular set; the $\mathcal L$-cut locus has zero contribution to the reduced-volume measure; the pre-cut regular set is carried smoothly by the unique minimizing $\mathcal L$-geodesic parametrization; this parametrization is invariant under the vector-field transport used below, so transported curves starting in the pre-cut region cannot hit the $\mathcal L$-cut locus before the target time; and the non-positive monotonicity density vanishes pointwise there. In particular, on the regular set of spacetime points reached by unique minimizing $\mathcal L$-geodesics before their cut time, the pointwise identity \begin{align*} \operatorname{Ric}_{g(\tau)} + \nabla^{g(\tau)}\nabla^{g(\tau)} l(\cdot,\tau) - \frac{1}{2\tau}g(\tau) = 0 \end{align*} holds. Here a regular basepoint means that the reduced exponential map is smooth away from its cut locus, the $\mathcal L$-cut locus is the first failure set of unique minimizing $\mathcal L$-geodesics, and the pre-cut regular set is the image of the reduced exponential map before that failure time. The hypotheses of this theorem are exactly the completeness, compact-time curvature bound, regular-basepoint, and finiteness assumptions in the statement. Since $\widetilde V$ is constant on $I$, its distributional derivative on every compact subinterval $J \subset I$ is zero. The equality case therefore gives, on the regular set of points reached by unique minimizing $\mathcal L$-geodesics before cut time, \begin{align*} \operatorname{Ric}_{g(\tau)} + \nabla^{g(\tau)}\nabla^{g(\tau)} l(\cdot,\tau) - \frac{1}{2\tau}g(\tau) = 0. \end{align*} Here $\nabla^{g(\tau)}\nabla^{g(\tau)} l(\cdot,\tau)$ denotes the Hessian of the function $x \mapsto l(x,\tau)$ with respect to $g(\tau)$.[/step]

[guided]Let $n := \dim M$. The quantity whose equality case we exploit is the reduced volume. The reduced-distance function is the map $l: M \times I \to \mathbb{R}$ based at the regular point $(p,0)$. For each time $\tau \in I$, define $v_\tau: M \to [0,\infty)$ by \begin{align*} v_\tau(x) = (4\pi\tau)^{-n/2} e^{-l(x,\tau)}. \end{align*} The reduced volume is the integral of this density against the Riemannian volume measure at time $\tau$: \begin{align*} \widetilde V(\tau) = \int_M v_\tau(x)\,d\mu_{g(\tau)}(x). \end{align*} The finiteness hypothesis ensures that this integral is a finite number. Completeness and bounded curvature on compact subintervals ensure that the reduced distance, minimizing $\mathcal L$-geodesics, and the monotonicity formula are available on compact time ranges. The regular-basepoint hypothesis supplies the standard regularity and cut-locus structure for $l$. We apply Perelman's reduced-volume monotonicity and equality-case rigidity theorem as the theorem-level equality-case input. Its hypotheses are: a complete backward Ricci flow, bounded curvature on compact time subintervals, a reduced distance based at a regular point, and finite reduced volume. These are exactly the hypotheses in the theorem statement. The argument below uses this result to obtain the tensor identity and then independently integrates that identity to recover the canonical shrinker form. In this statement, a regular basepoint means that the reduced exponential map has the usual smooth pre-cut parametrization, the $\mathcal L$-cut locus is the first set where uniqueness or smoothness of minimizing $\mathcal L$-geodesics fails, and the pre-cut regular set is the image before that cut time. The theorem concludes first that $\widetilde V$ is non-increasing. Its equality case says that if the distributional derivative of $\widetilde V$ vanishes on a compact time interval, then the non-positive monotonicity density vanishes on the regular part of spacetime, the reduced distance is smooth there, the $\mathcal L$-cut locus has zero reduced-volume measure, the unique minimizing $\mathcal L$-geodesic parametrization persists smoothly on the pre-cut region, and the vector-field transport used in the soliton reconstruction preserves that pre-cut region up to the target time. Therefore the Hessian identity below holds pointwise exactly on the regular generated region where the tensor is defined. Since $\widetilde V$ is constant on $I$, its distributional derivative vanishes on every compact subinterval $J \subset I$. Therefore the equality-case conclusion applies and yields \begin{align*} \operatorname{Ric}_{g(\tau)} + \nabla^{g(\tau)}\nabla^{g(\tau)} l(\cdot,\tau) - \frac{1}{2\tau}g(\tau) = 0 \end{align*} on the set where $l$ is smooth, equivalently on the regular set reached by unique minimizing $\mathcal L$-geodesics from the basepoint before cut time. This is the key tensor identity: it is the gradient shrinking Ricci soliton equation written in backward time with potential $l(\cdot,\tau)$.[/guided]

[step:Restrict to the regular generated region where the reduced distance is smooth] Let $\mathcal G \subset M \times I$ be the regular generated region, defined as the set of spacetime points reached from the basepoint by a unique minimizing $\mathcal L$-geodesic before its cut time. For each $\tau \in I$, let $\mathcal G_\tau := \{x \in M : (x,\tau) \in \mathcal G\}$, called the minimizing-generated time slice. By the regular-basepoint and cut-locus conclusions in Perelman's equality-case theorem, $\mathcal G$ is an open smooth spacetime region, $l$ is smooth on $\mathcal G$, and the non-regular part consisting of cut-locus points has zero reduced-volume measure $v_\tau\,d\mu_{g(\tau)}$ in each time slice. Thus the cut locus is used only to justify that no positive reduced-volume contribution is missed in the monotonicity formula; we do not assert a pointwise Hessian identity at cut-locus points where $l$ may fail to be smooth. Therefore the tensor identity obtained in the previous step holds pointwise on $\mathcal G$: \begin{align*} \operatorname{Ric}_{g(\tau)} + \nabla^{g(\tau)}\nabla^{g(\tau)} l(\cdot,\tau) - \frac{1}{2\tau}g(\tau) = 0. \end{align*} Connectedness of $M$ is used only to phrase the ambient flow as one connected object and to avoid a componentwise theorem statement; the rigidity argument is componentwise. The soliton reconstruction itself is performed on each connected component of the regular generated region, and no step below uses paths through cut-locus points to connect distinct components. [/step]

[step:Reconstruct the canonical shrinking soliton flow from the tensor identity]Fix a time $\tau_1 \in I$ and define the potential $f: \mathcal G_{\tau_1} \to \mathbb{R}$ by \begin{align*} f(x) = l(x,\tau_1). \end{align*} The tensor identity at time $\tau_1$ gives \begin{align*} \operatorname{Ric}_{g(\tau_1)} + \nabla^{g(\tau_1)}\nabla^{g(\tau_1)} f = \frac{1}{2\tau_1}g(\tau_1), \end{align*} so $(\mathcal G_{\tau_1},g(\tau_1),f)$ is a gradient shrinking Ricci soliton on the regular generated region. Let $X_\tau: \mathcal G_\tau \to T\mathcal G_\tau$ be the time-dependent vector field defined by \begin{align*} X_\tau(x) = \nabla^{g(\tau)} l(x,\tau). \end{align*} The domain-transport clause in Perelman's equality-case rigidity theorem supplies the required domain statement: the integral curves of $X_\tau$ that start in $\mathcal G_{\tau_1}$ exist for all $\tau \in I$, remain in the regular generated region, cannot hit the $\mathcal L$-cut locus before the target time, and define diffeomorphisms $\Phi_{\tau_1,\tau}: \mathcal G_{\tau_1} \to \mathcal G_\tau$. Equivalently, these maps are the pre-cut reduced-exponential parametrizations transported along the same minimizing $\mathcal L$-geodesics, so no point of the stated regular generated slice is lost. They satisfy \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}(x) = X_\tau(\Phi_{\tau_1,\tau}(x)) \end{align*} and \begin{align*} \Phi_{\tau_1,\tau_1}(x)=x. \end{align*} Using the backward Ricci flow equation $\partial_\tau g(\tau)=2\operatorname{Ric}_{g(\tau)}$ and the identity $\mathcal L_{\nabla^{g(\tau)}l}g(\tau)=2\nabla^{g(\tau)}\nabla^{g(\tau)}l(\cdot,\tau)$, we first get \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \Phi_{\tau_1,\tau}^*\left(\partial_\tau g(\tau)+\mathcal L_{X_\tau}g(\tau)\right). \end{align*} Substituting the backward Ricci flow equation and the Lie derivative identity gives \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \Phi_{\tau_1,\tau}^*\left(2\operatorname{Ric}_{g(\tau)}+2\nabla^{g(\tau)}\nabla^{g(\tau)}l(\cdot,\tau)\right). \end{align*} Using the tensor identity from the equality case, this becomes \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \frac{1}{\tau}\Phi_{\tau_1,\tau}^*g(\tau). \end{align*} Solving this scalar linear equation for the pulled-back metric gives \begin{align*} \Phi_{\tau_1,\tau}^*g(\tau) = \frac{\tau}{\tau_1} g(\tau_1). \end{align*} Thus, after pullback by diffeomorphisms and parabolic rescaling by $\tau_1/\tau$, the flow on each connected component of $\mathcal G$ is the canonical backward flow generated by the gradient shrinking Ricci soliton $(\mathcal G_{\tau_1},g(\tau_1),f)$.[/step]

[guided]We now reconstruct the soliton form from the tensor identity. Fix $\tau_1 \in I$ and define $f: \mathcal G_{\tau_1} \to \mathbb{R}$ by \begin{align*} f(x) = l(x,\tau_1). \end{align*} Because $l$ is smooth on the regular generated region, this is a smooth potential on $\mathcal G_{\tau_1}$. At time $\tau_1$, the equality-case identity becomes \begin{align*} \operatorname{Ric}_{g(\tau_1)} + \nabla^{g(\tau_1)}\nabla^{g(\tau_1)} f = \frac{1}{2\tau_1}g(\tau_1). \end{align*} This is precisely the gradient shrinking Ricci soliton equation with shrinker scale $\tau_1$. The correct soliton-generating vector field is not $\tau^{-1}\nabla l$. Define $X_\tau: \mathcal G_\tau \to T\mathcal G_\tau$ by \begin{align*} X_\tau(x) = \nabla^{g(\tau)}l(x,\tau). \end{align*} There is a domain issue here because $\mathcal G$ is open and bounded by the $\mathcal L$-cut locus. The domain-transport clause in Perelman's equality-case rigidity theorem supplies the needed conclusion: the integral curves of $X_\tau$ starting at points of $\mathcal G_{\tau_1}$ exist for every $\tau \in I$, stay inside $\mathcal G$, cannot hit the $\mathcal L$-cut locus before the target time, and give diffeomorphisms $\Phi_{\tau_1,\tau}: \mathcal G_{\tau_1} \to \mathcal G_\tau$. Equivalently, the maps transport the pre-cut reduced-exponential parametrization along the same unique minimizing $\mathcal L$-geodesics. Hence they cover exactly the stated regular generated time slices and satisfy \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}(x) = X_\tau(\Phi_{\tau_1,\tau}(x)) \end{align*} and \begin{align*} \Phi_{\tau_1,\tau_1}(x)=x. \end{align*} The metric derivative along a time-dependent pullback is \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \Phi_{\tau_1,\tau}^*\left(\partial_\tau g(\tau)+\mathcal L_{X_\tau}g(\tau)\right). \end{align*} For a backward Ricci flow, $\partial_\tau g(\tau)=2\operatorname{Ric}_{g(\tau)}$. Since $X_\tau=\nabla^{g(\tau)}l(\cdot,\tau)$, the Lie derivative identity for a gradient vector field gives \begin{align*} \mathcal L_{X_\tau}g(\tau)=2\nabla^{g(\tau)}\nabla^{g(\tau)}l(\cdot,\tau). \end{align*} Substituting these two identities gives \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \Phi_{\tau_1,\tau}^*\left(2\operatorname{Ric}_{g(\tau)}+2\nabla^{g(\tau)}\nabla^{g(\tau)}l(\cdot,\tau)\right). \end{align*} The soliton tensor identity then yields \begin{align*} \frac{d}{d\tau}\Phi_{\tau_1,\tau}^*g(\tau) = \frac{1}{\tau}\Phi_{\tau_1,\tau}^*g(\tau). \end{align*} This ordinary differential equation in the space of symmetric two-tensors has initial value $\Phi_{\tau_1,\tau_1}^*g(\tau_1)=g(\tau_1)$, hence \begin{align*} \Phi_{\tau_1,\tau}^*g(\tau)=\frac{\tau}{\tau_1}g(\tau_1). \end{align*} Therefore the pulled-back flow differs from the fixed soliton metric only by the parabolic scale factor $\tau/\tau_1$, which is exactly the canonical backward shrinker form.[/guided]

[step:Verify constant reduced volume for a canonical normalized shrinker]Conversely, let $(N^n,h,f)$ be a complete gradient shrinking Ricci soliton with normalized potential $f: N \to \mathbb{R}$, so \begin{align*} \operatorname{Ric}_h + \nabla^h\nabla^h f = \frac{1}{2}h. \end{align*} Here normalized means precisely the normalization in the canonical-flow theorem for complete gradient shrinking Ricci solitons: the Gaussian density is finite and the reduced distance for the canonical flow based at the soliton basepoint is represented by $f$ after undoing the canonical diffeomorphism. Let $\Psi_\tau: N \to N$ be the canonical diffeomorphisms generated by the soliton vector field, written with the convention that $\Psi_\tau$ sends the fixed soliton coordinate $x$ to the corresponding point at backward time $\tau$. Define the canonical backward flow by \begin{align*} g(\tau) := \tau\,(\Psi_\tau^{-1})^*h. \end{align*} By that canonical-flow theorem, completeness makes the soliton vector field complete, the maps $\Psi_\tau$ are global diffeomorphisms on the time interval, and the reduced distance based at the soliton basepoint satisfies \begin{align*} l(\Psi_\tau(x),\tau) = f(x). \end{align*} The Riemannian volume measure transforms under the pushforward by $\Psi_\tau$ and parabolic rescaling by \begin{align*} d\mu_{g(\tau)}(\Psi_\tau(x)) = \tau^{n/2}\,d\mu_h(x). \end{align*} Therefore, after the substitution $y=\Psi_\tau(x)$, the reduced volume is \begin{align*} \widetilde V(\tau) = \int_N (4\pi\tau)^{-n/2}e^{-l(y,\tau)}\,d\mu_{g(\tau)}(y). \end{align*} Using $l(\Psi_\tau(x),\tau)=f(x)$ and $d\mu_{g(\tau)}(\Psi_\tau(x))=\tau^{n/2}\,d\mu_h(x)$ gives \begin{align*} \widetilde V(\tau) = \int_N (4\pi\tau)^{-n/2}e^{-f(x)}\tau^{n/2}\,d\mu_h(x). \end{align*} Cancelling the powers of $\tau$ yields \begin{align*} \widetilde V(\tau) = \int_N (4\pi)^{-n/2}e^{-f(x)}\,d\mu_h(x). \end{align*} The last expression is independent of $\tau$. Hence the reduced volume of the canonical backward flow of a complete normalized gradient shrinking Ricci soliton is constant on its time interval.[/step]

[guided]For the converse, start with a complete gradient shrinking Ricci soliton $(N^n,h,f)$, where $f: N \to \mathbb{R}$ is a normalized potential and \begin{align*} \operatorname{Ric}_h + \nabla^h\nabla^h f = \frac{1}{2}h. \end{align*} The normalization is the standard shrinker normalization for which the Gaussian density is finite, \begin{align*} \int_N (4\pi)^{-n/2}e^{-f(x)}\,d\mu_h(x) < \infty, \end{align*} and the reduced distance in the canonical backward flow pulls back to $f$. This is not a consequence we leave implicit: we invoke the canonical-flow theorem for complete gradient shrinking Ricci solitons. That theorem says that completeness makes the soliton vector field complete and produces global diffeomorphisms $\Psi_\tau: N \to N$. We use the convention that $\Psi_\tau$ sends the fixed soliton coordinate $x$ to the corresponding point at backward time $\tau$, so the complete backward Ricci flow is \begin{align*} g(\tau)=\tau\,(\Psi_\tau^{-1})^*h, \end{align*} and, with the stated normalization of $f$, the reduced distance is identified by \begin{align*} l(\Psi_\tau(x),\tau)=f(x). \end{align*} The factor $\tau$ in the metric rescales $n$-dimensional Riemannian volume by $\tau^{n/2}$, while the diffeomorphism convention means that the volume measure at the point $\Psi_\tau(x)$ is the pushforward of this rescaled soliton volume. Thus \begin{align*} d\mu_{g(\tau)}(\Psi_\tau(x))=\tau^{n/2}\,d\mu_h(x). \end{align*} Substituting $y=\Psi_\tau(x)$ in the reduced-volume integral gives \begin{align*} \widetilde V(\tau) = \int_N (4\pi\tau)^{-n/2}e^{-l(y,\tau)}\,d\mu_{g(\tau)}(y). \end{align*} The identities $l(\Psi_\tau(x),\tau)=f(x)$ and $d\mu_{g(\tau)}(\Psi_\tau(x))=\tau^{n/2}\,d\mu_h(x)$ transform this into \begin{align*} \widetilde V(\tau) = \int_N (4\pi\tau)^{-n/2}e^{-f(x)}\tau^{n/2}\,d\mu_h(x). \end{align*} Cancelling the scale factors gives \begin{align*} \widetilde V(\tau) = \int_N (4\pi)^{-n/2}e^{-f(x)}\,d\mu_h(x). \end{align*} All dependence on $\tau$ cancels. Therefore $\widetilde V$ is constant on the time interval of the canonical backward flow.[/guided]